UPDATE: after checking, we found that the Spaces activation provides the DB error connected with the Timeline. So we've upgraded all modules via the Apps Market and launched the query that clears all mentions of the Spaces in the alert system:

delete FROM `bx_timeline_handlers` WHERE `group` like 'bx_spaces%';

delete FROM `sys_alerts` WHERE `unit` like 'bx_spaces%';

delete FROM `bx_notifications_handlers` WHERE `alert_unit` like 'bx_spaces%';

and activated Spaces again. After that, the Spaces app was activated successfully.