@Ray
About the only thing that comes to mind is automated backups. If on, turn to either manual or off.
This gets to be kinda tricky if one cannot get to the admin interface of the site and the automated backups were saved to the default area which is that sea of hashed files in moodledata/filedir/
Need to free up enough space to be able to login ... so with whatever you have to browse files on the server, you need to manually remove the files/folders in the following areas:
moodledata/sessions/ - those are small files and won't gain much space but they are created upon login
moodledata/trashdir/ - all the contents
moodledata/temp/backup/ - this is the area used by moodle to build backups and can be manually removed without any issues.
Hopefully, that will free up enough space so you can login as admin user and disable autobackups.
Then the next step ... go into courses ... hit restore link ... you are not going to restore any course, just trying to get to some Moodle screen that would allow you to remove automated backup files.
Suggest keeping one backup ... but definitely download the most recent backup of the course.
Hopefully, the above will give you a place to start!
'spirit of sharing', Ken