The error reads like this:

Fatal Error.

DELAYED option not supported for table 'elgg_system_log'
QUERY: INSERT DELAYED into elgg_system_log (object_id, object_class, object_type, object_subtype, event, performed_by_guid, owner_guid, access_id, enabled, time_created, ip_address) VALUES ('331','ElggPlugin','object', 'plugin', 'disable', 35, 1, 2, 'no', '1359626211', '80.193.98.17')

DatabaseException Object ( [message:protected] => DELAYED option not supported for table 'elgg_system_log'
QUERY: INSERT DELAYED into elgg_system_log (object_id, object_class, object_type, object_subtype, event, performed_by_guid, owner_guid, access_id, enabled, time_created, ip_address) VALUES ('331','ElggPlugin','object', 'plugin', 'disable', 35, 1, 2, 'no', '1359626211', '80.193.98.17') [string:Exception:private] => exception 'DatabaseException' with message 'DELAYED option not supported for table 'elgg_system_log'
QUERY: INSERT DELAYED into elgg_system_log (object_id, object_class, object_type, object_subtype, event, performed_by_guid, owner_guid, access_id, enabled, time_created, ip_address) VALUES ('331','ElggPlugin','object', 'plugin', 'disable', 35, 1, 2, 'no', '1359626211', '80.193.98.17')' in /home/sites/kalulushi.org/public_html/staff/engine/lib/database.php:265 Stack trace: #0 /home/sites/kalulushi.org/public_html/staff/engine/lib/database.php(467): execute_query('INSERT DELAYED ...', Resource id #55) #1 /home/sites/kalulushi.org/public_html/staff/engine/lib/system_log.php(216): insert_data('INSERT DELAYED ...') #2 /home/sites/kalulushi.org/public_html/staff/engine/lib/system_log.php(271): system_log(Object(ElggPlugin), 'disable') #3 [internal function]: system_log_default_logger('log', 'systemlog', Array) #4 /home/sites/kalulushi.org/public_html/staff/engine/lib/elgglib.php(776): call_user_func_array('system_log_defa...', Array) #5 /home/sites/kalulushi.org/public_html/staff/engine/lib/system_log.php(289): elgg_trigger_event('log', 'systemlog', Array) #6 [internal function]: system_log_listener('disable', 'object', Object(ElggPlugin)) #7 /home/sites/kalulushi.org/public_html/staff/engine/lib/elgglib.php(776): call_user_func_array('system_log_list...', Array) #8 /home/sites/kalulushi.org/public_html/staff/engine/lib/entities.php(1586): elgg_trigger_event('disable', 'object', Object(ElggPlugin)) #9 /home/sites/kalulushi.org/public_html/staff/engine/classes/ElggEntity.php(1393): disable_entity(331, '', true) #10 /home/sites/kalulushi.org/public_html/staff/engine/lib/plugins.php(167): ElggEntity->disable() #11 /home/sites/kalulushi.org/public_html/staff/views/default/admin/plugins.php(14): elgg_generate_plugin_entities() #12 /home/sites/kalulushi.org/public_html/staff/engine/lib/views.php(491): include('/home/sites/kal...') #13 /home/sites/kalulushi.org/public_html/staff/engine/lib/admin.php(486): elgg_view('admin/plugins', Array) #14 [internal function]: admin_page_handler(Array, 'admin') #15 /home/sites/kalulushi.org/public_html/staff/engine/lib/pagehandler.php(53): call_user_func('admin_page_hand...', Array, 'admin') #16 /home/sites/kalulushi.org/public_html/staff/engine/handlers/page_handler.php(46): page_handler('admin', 'plugins') #17 {main} [code:protected] => 0 [file:protected] => /home/sites/kalulushi.org/public_html/staff/engine/lib/database.php [line:protected] => 265 [trace:Exception:private] => Array ( [0] => Array ( [file] => /home/sites/kalulushi.org/public_html/staff/engine/lib/database.php [line] => 467 [function] => execute_query [args] => Array ( [0] => INSERT DELAYED into elgg_system_log (object_id, object_class, object_type, object_subtype, event, performed_by_guid, owner_guid, access_id, enabled, time_created, ip_address) VALUES ('331','ElggPlugin','object', 'plugin', 'disable', 35, 1, 2, 'no', '1359626211', '80.193.98.17') [1] => Resource id #55 )
)
[1] => Array ( [file] => /home/sites/kalulushi.org/public_html/staff/engine/lib/system_log.php [line] => 216 [function] => insert_data [args] => Array ( [0] => INSERT DELAYED into elgg_system_log (object_id, object_class, object_type, object_subtype, event, performed_by_guid, owner_guid, access_id, enabled, time_created, ip_address) VALUES ('331','ElggPlugin','object', 'plugin', 'disable', 35, 1, 2, 'no', '1359626211', '80.193.98.17') )
)
[2] => Array ( [file] => /home/sites/kalulushi.org/public_html/staff/engine/lib/system_log.php [line] => 271 [function] => system_log [args] => Array ( [0] => ElggPlugin Object ( [package:ElggPlugin:private] =>
[manifest:ElggPlugin:private] => [path:ElggPlugin:private] => /home/sites/kalulushi.org/public_html/staff/mod/Elgg-Event-Calendar-full [pluginID:ElggPlugin:private] => [errorMsg:ElggPlugin:private] => [url_override:protected] =>
[icon_override:protected] => [temp_metadata:protected] => Array ( )
[temp_annotations:protected] => Array ( )
[temp_private_settings:protected] => Array ( )
[volatile:protected] => Array ( )
[attributes:protected] => Array ( [guid] => 331 [type] => object [subtype] => 2 [owner_guid] => 1 [site_guid] => 1 [container_guid] => 1 [access_id] => 2 [time_created] => 1359625636 [time_updated] => 1359625636 [last_action] => 1359625636 [enabled] => no [title] => Elgg-Event-Calendar-full [description] => [tables_split] => 2 [tables_loaded] => 2 )
[valid:protected] =>
)
[1] => disable )
)
[3] => Array ( [function] => system_log_default_logger [args] => Array ( [0] => log [1] => systemlog [2] => Array ( [object] => ElggPlugin Object ( [package:ElggPlugin:private] =>
[manifest:ElggPlugin:private] => [path:ElggPlugin:private] => /home/sites/kalulushi.org/public_html/staff/mod/Elgg-Event-Calendar-full [pluginID:ElggPlugin:private] => [errorMsg:ElggPlugin:private] => [url_override:protected] =>
[icon_override:protected] => [temp_metadata:protected] => Array ( )
[temp_annotations:protected] => Array ( )
[temp_private_settings:protected] => Array ( )
[volatile:protected] => Array ( )
[attributes:protected] => Array ( [guid] => 331 [type] => object [subtype] => 2 [owner_guid] => 1 [site_guid] => 1 [container_guid] => 1 [access_id] => 2 [time_created] => 1359625636 [time_updated] => 1359625636 [last_action] => 1359625636 [enabled] => no [title] => Elgg-Event-Calendar-full [description] => [tables_split] => 2 [tables_loaded] => 2 )
[valid:protected] =>
)
[event] => disable )
)
)
[4] => Array ( [file] => /home/sites/kalulushi.org/public_html/staff/engine/lib/elgglib.php [line] => 776 [function] => call_user_func_array [args] => Array ( [0] => system_log_default_logger [1] => Array ( [0] => log [1] => systemlog [2] => Array ( [object] => ElggPlugin Object ( [package:ElggPlugin:private] =>
[manifest:ElggPlugin:private] => [path:ElggPlugin:private] => /home/sites/kalulushi.org/public_html/staff/mod/Elgg-Event-Calendar-full [pluginID:ElggPlugin:private] => [errorMsg:ElggPlugin:private] => [url_override:protected] =>
[icon_override:protected] => [temp_metadata:protected] => Array ( )
[temp_annotations:protected] => Array ( )
[temp_private_settings:protected] => Array ( )
[volatile:protected] => Array ( )
[attributes:protected] => Array ( [guid] => 331 [type] => object [subtype] => 2 [owner_guid] => 1 [site_guid] => 1 [container_guid] => 1 [access_id] => 2 [time_created] => 1359625636 [time_updated] => 1359625636 [last_action] => 1359625636 [enabled] => no [title] => Elgg-Event-Calendar-full [description] => [tables_split] => 2 [tables_loaded] => 2 )
[valid:protected] =>
)
[event] => disable )
)
)
)
[5] => Array ( [file] => /home/sites/kalulushi.org/public_html/staff/engine/lib/system_log.php [line] => 289 [function] => elgg_trigger_event [args] => Array ( [0] => log [1] => systemlog [2] => Array ( [object] => ElggPlugin Object ( [package:ElggPlugin:private] =>
[manifest:ElggPlugin:private] => [path:ElggPlugin:private] => /home/sites/kalulushi.org/public_html/staff/mod/Elgg-Event-Calendar-full [pluginID:ElggPlugin:private] => [errorMsg:ElggPlugin:private] => [url_override:protected] =>
[icon_override:protected] => [temp_metadata:protected] => Array ( )
[temp_annotations:protected] => Array ( )
[temp_private_settings:protected] => Array ( )
[volatile:protected] => Array ( )
[attributes:protected] => Array ( [guid] => 331 [type] => object [subtype] => 2 [owner_guid] => 1 [site_guid] => 1 [container_guid] => 1 [access_id] => 2 [time_created] => 1359625636 [time_updated] => 1359625636 [last_action] => 1359625636 [enabled] => no [title] => Elgg-Event-Calendar-full [description] => [tables_split] => 2 [tables_loaded] => 2 )
[valid:protected] =>
)
[event] => disable )
)
)
[6] => Array ( [function] => system_log_listener [args] => Array ( [0] => disable [1] => object [2] => ElggPlugin Object ( [package:ElggPlugin:private] =>
[manifest:ElggPlugin:private] => [path:ElggPlugin:private] => /home/sites/kalulushi.org/public_html/staff/mod/Elgg-Event-Calendar-full [pluginID:ElggPlugin:private] => [errorMsg:ElggPlugin:private] => [url_override:protected] =>
[icon_override:protected] => [temp_metadata:protected] => Array ( )
[temp_annotations:protected] => Array ( )
[temp_private_settings:protected] => Array ( )
[volatile:protected] => Array ( )
[attributes:protected] => Array ( [guid] => 331 [type] => object [subtype] => 2 [owner_guid] => 1 [site_guid] => 1 [container_guid] => 1 [access_id] => 2 [time_created] => 1359625636 [time_updated] => 1359625636 [last_action] => 1359625636 [enabled] => no [title] => Elgg-Event-Calendar-full [description] => [tables_split] => 2 [tables_loaded] => 2 )
[valid:protected] =>
)
)
)
[7] => Array ( [file] => /home/sites/kalulushi.org/public_html/staff/engine/lib/elgglib.php [line] => 776 [function] => call_user_func_array [args] => Array ( [0] => system_log_listener [1] => Array ( [0] => disable [1] => object [2] => ElggPlugin Object ( [package:ElggPlugin:private] =>
[manifest:ElggPlugin:private] => [path:ElggPlugin:private] => /home/sites/kalulushi.org/public_html/staff/mod/Elgg-Event-Calendar-full [pluginID:ElggPlugin:private] => [errorMsg:ElggPlugin:private] => [url_override:protected] =>
[icon_override:protected] => [temp_metadata:protected] => Array ( )
[temp_annotations:protected] => Array ( )
[temp_private_settings:protected] => Array ( )
[volatile:protected] => Array ( )
[attributes:protected] => Array ( [guid] => 331 [type] => object [subtype] => 2 [owner_guid] => 1 [site_guid] => 1 [container_guid] => 1 [access_id] => 2 [time_created] => 1359625636 [time_updated] => 1359625636 [last_action] => 1359625636 [enabled] => no [title] => Elgg-Event-Calendar-full [description] => [tables_split] => 2 [tables_loaded] => 2 )
[valid:protected] =>
)
)
)
)
[8] => Array ( [file] => /home/sites/kalulushi.org/public_html/staff/engine/lib/entities.php [line] => 1586 [function] => elgg_trigger_event [args] => Array ( [0] => disable [1] => object [2] => ElggPlugin Object ( [package:ElggPlugin:private] =>
[manifest:ElggPlugin:private] => [path:ElggPlugin:private] => /home/sites/kalulushi.org/public_html/staff/mod/Elgg-Event-Calendar-full [pluginID:ElggPlugin:private] => [errorMsg:ElggPlugin:private] => [url_override:protected] =>
[icon_override:protected] => [temp_metadata:protected] => Array ( )
[temp_annotations:protected] => Array ( )
[temp_private_settings:protected] => Array ( )
[volatile:protected] => Array ( )
[attributes:protected] => Array ( [guid] => 331 [type] => object [subtype] => 2 [owner_guid] => 1 [site_guid] => 1 [container_guid] => 1 [access_id] => 2 [time_created] => 1359625636 [time_updated] => 1359625636 [last_action] => 1359625636 [enabled] => no [title] => Elgg-Event-Calendar-full [description] => [tables_split] => 2 [tables_loaded] => 2 )
[valid:protected] =>
)
)
)
[9] => Array ( [file] => /home/sites/kalulushi.org/public_html/staff/engine/classes/ElggEntity.php [line] => 1393 [function] => disable_entity [args] => Array ( [0] => 331 [1] => [2] => 1 )
)
[10] => Array ( [file] => /home/sites/kalulushi.org/public_html/staff/engine/lib/plugins.php [line] => 167 [function] => disable [class] => ElggEntity [type] => -> [args] => Array ( )
)
[11] => Array ( [file] => /home/sites/kalulushi.org/public_html/staff/views/default/admin/plugins.php [line] => 14 [function] => elgg_generate_plugin_entities [args] => Array ( )
)
[12] => Array ( [file] => /home/sites/kalulushi.org/public_html/staff/engine/lib/views.php [line] => 491 [args] => Array ( [0] => /home/sites/kalulushi.org/public_html/staff/views/default/admin/plugins.php )
[function] => include )
[13] => Array ( [file] => /home/sites/kalulushi.org/public_html/staff/engine/lib/admin.php [line] => 486 [function] => elgg_view [args] => Array ( [0] => admin/plugins [1] => Array ( [page] => Array ( [0] => plugins )
)
)
)
[14] => Array ( [function] => admin_page_handler [args] => Array ( [0] => Array ( [0] => plugins )
[1] => admin )
)
[15] => Array ( [file] => /home/sites/kalulushi.org/public_html/staff/engine/lib/pagehandler.php [line] => 53 [function] => call_user_func [args] => Array ( [0] => admin_page_handler [1] => Array ( [0] => plugins )
[2] => admin )
)
[16] => Array ( [file] => /home/sites/kalulushi.org/public_html/staff/engine/handlers/page_handler.php [line] => 46 [function] => page_handler [args] => Array ( [0] => admin [1] => plugins )
)
)
[previous:Exception:private] => )

such a longgggggggggg post of your error message when only the first 3 lines are neeeded !
probem is not an elgg issue. 
someone must have got smart with the system log table
and changed the table storage type.
need to reverse whatever you've done to that table ;-P
put it back to myisam.. ?!;-P

yeah you're gonna have to contact your host or restore your elgg back to before 1.8.13. this is an error due to your mysql table type isn't in MySIAM. did someone convert it to innoDB or something else?

He could convert the table back to MyISAM - doesn't need to restore the DB.

http://www.kavoir.com/2009/09/mysql-how-to-change-or-convert-myisam-to-innodb-or-vice-versa.html

I'm the only administrator with access.  It may have been my hosts, as the site was down briefly yesterday.

I'll try that.  Thanks, guys.

Works a treat.  Converted the offending tables to MyISAM with phpmyadmin and it's fixed.  You lot are beyond genius.  Thank you so much!

@cash nice.

@craig np

wonder which stranger nicely said ' put it back to myisam.. ?!;-P ' ? n/p