All Plesk “local” backups are stored in the following path:
The problem resides in the fact that if the partition containing /var is small, this can create big problems. Luckily Plesk is happy with symlinks…
So in order to raise the space available for dumps, simply move the content of the dumps directory to the new location and create a symlink:
mv dumps /home/backups/
ln -s /home/backups/dumps dumps
This creates a symbolic link to the new dump path which has plenty of disk space.