Built-in Postgres' Automatic Backup Is there any automatic built-in mechanism in postgres to backup data if the data have exceeded the maximum permissible storage amount?
The only maximum permissible storage amount is the amount of storage your disk has. When you're running out of space, postgreSQL will simply stop. (with a PANIC message)

You have to monitor your disks free space to prevent it to happen...

Sadly documentation is quite light on it:
https://www.postgresql.org/docs/9.6/static/diskusage.html