My first response- Did you "Choose Different Backup Folder..." ? This would allow you to save to a drive location that is available such as an internal drive.
Second- This is only a 'Backup', and would your working 'master' catalog be safe on a drive location that is available? Just reopen LR and exit again creating a current backup.
Thirdly- I hope you are backing up to a physical drive that is different to the drive with the 'master' catalog.!
Are you saying that the hard-drive where your LR catalog resides got unplugged and is no longer available to LR and when you try to shut down LR you're getting an error when trying to backup on the way out?
In other words have you not exited LR after turning off the catalog's hard-drive? I would expect LR to be confused if it lost contact with the catalog hard-drive in the middle of an editing session and it may not do anything useful until you restart it. Of course the catalog may have issues once you do restart it. If you are choosing a backup location that has enough space, then I'd guess LR cannot access its catalog anymore and has nothing it can back up. I think the only thing you'll be able to do is Quit.
If your situation isn't' that you turned off or disconnected the catalog hard-drive then please describe more specifically how the drive you disconnected or turned off is related to the LR catalog and/or LR program.