If you're on a shared environment, then I'd suggest perhaps reviewing who you host with. Do you have a normal Cpanel (or other panel) type of account or do you have a VPS (with root access) ? If you're on a VPS, then it's your job to manage this, if not, then your host should take care of this.

The file system will fill up if it's not properly managed or is sized incorrectly. There's possibly log files being written there and no purge routine. Without seeing the system it's not possible to suggest the exact cause.

Generally a full file system is a sign of pending doom