Everything breaks at once and nothing indicates why.
CONFIRM IT
df -h. A filesystem at 100 per cent explains almost any symptom.
Also check inodes with df -i, which can be exhausted while space remains.
FREE SPACE QUICKLY
Safe first candidates: Old log files and rotated archives Package manager caches Local backup files, if copies exist elsewhere Temporary files
Do not delete things you do not recognise. Deleting the wrong file makes recovery harder.
FIND WHAT GREW
du, working down from the root, largest first.
WATCH FOR DELETED-BUT-OPEN FILES
A deleted file held open by a process does not release its space until that process restarts. df shows the disk full while du cannot find the data. Restart the relevant service.
AFTER RECOVERING
Work out why it filled. Unrotated logs and local backups are the usual answers.
Configure rotation, move backups off the server, and set an alert at eighty per cent so you are warned rather than surprised.