This looks like an issue starting the session.
By default sessions use the file store and are kept in the sessions folder in Moodledata. Check the disk or filesystem containing Moodledata isn't full.
If the site isn't using the file store check the $CFG->session_handler_class setting in config.php in the site's source code folder. This should point to what to do next.
It's possible there's another configuration setting causing sessions to be stored in the database but I'd expect a different error.