Ok,
I think that at least one partition is full or something like that. Do you have a chance to login to the server (ssh, telnet ...) ? If yes, then log in and execute the command
df -m
That should return a table with all partitions (mount points) and info about total/used/free spache in MB.
I think you need to get in contact with your hoster.
Another possible reason: Do you have very large tables so that one of the database file might be hitting a 2GB limit or something like that ?
Thomas