What happens when a hard drive gets full with Linux running? Does it lock the system? Or something else, or nothing happens?
I am using Ubuntu Linux.
It should not hang the system; however, applications will fail when they try to write to files and log files might not get updated.
In ext filesystems, an amount of disk is reserved to the root user; this might suit your use case (you don't want your system to have a messy failure by being unable to write a log file).
In ext2/ext3/ext4 file systems 5% of disk space is reserved for root in case of disk being full so processes can work properly. You can check this with command:
Where /dev/sda1 is device of filesystem (you can check it with df command).
You can alter this value with tune2fs command: