If you can access config.php file and edit, one could possibly turn on auth via email by adding a line to config.php. If you have other authentications, like LDAP/other, you should include those as well or you'll then lock those users out.
Manual accounts are always on.
Students using mobile devices and the Moodle App? then one would also need 'webservices' added to the line.
Lines added just *above* the comment at the end of the config.php file - not below that comment. No need to restart services, just hit site with browser.
That then gets you in where you could navigate to the GUI area for turning on and set that there.
After everything 'back to normal', comment out the lines added by placing // in front of them.