There's no benefit to fixing a bad drive. Hard disks are very inexpensive and I wouldn't want to trust my data to a repaired hard disk. As for backing up the time to do that is before the drive fails. You're now looking at a recovery effort.
Depending on the nature of the failure you may be able to rebuild the system on a new drive, connect the damaged drive, and see if the OS / Disk Utility will recognize it. If you're lucky it will and you can copy whatever you can off the drive. If not then you can investigate recovery software. I've used some Windows based recovery software (to recover information on a Windows drive) and it worked reasonably well (assuming the drive hasn't completely failed). I do not know what options exist for the Mac. And finally, if the information is really important you can send the drive out for recovery. Costs can vary but $1K - $3K is what I've heard the PC support guys tell employees when they need to do this.
Good luck.