The "disk space full" message is indicating that the actual storage space shared by all of the users on your server (and by some of the server processes as well) is nearly full. It does not indicate that your account is at/near/over its limit. The server admins will need to search for the files or accounts that may be causing the problem (things like temporary files that need to be deleted, error logs that have become huge because of a programming mistake, people uploading large files that violate the terms of service, that sort of thing). Until the problems are sorted out, writing files to the system is disabled to prevent corruption of existing and new files.
It may take some time to clear up the problem, so please be patient.