Well, two things: first, it would probably be a good idea to back your data up now if you can. You can probably go ahead and fix the MBR anyway. But, should something go wrong that renders the disk unusable, you might wish you had it backed up.
But getting the data off of a drive with a dead operating system takes some doing. You obviously have a working computer, but what sort of equipment are we dealing with? Do you have a thumb drive? Something like this.
If so, that's one step in the right direction.
The next step is, do you have a CD-burner? Can you burn an ISO image?
If you can, then you can get a disk containing an a small, bootable operating system that will allow you to go in and using clumsy filesystem managers you can move files from the hard drive to the thumb drive, then to the other computer.
One disk that I use is UBCD
. It has a lot of tools on it, and boots from the CD-Rom. If the PC will boot with this, you can copy the files without doing anything at all to the drive.
Another option is to remove the drive from the computer, and (assuming it's an IDE drive) switch the jumper to "slave" from "master", then put it in another computer. You can then copy the files from one drive to another, assuming there's space on the host drive.
All that said, regarding the MBR fix, read here
Keep in mind that anything you do to your hard drive can potentially destroy it. That includes using it. :-)
And it's always possible that the drive is failing, but there's no way to know until you do some work. Handle it by the sides if you move it, don't touch the exposed electronics if possible. Don't shuffle across carpet and build up electrostatic electricity. And in future, keep this one fact in mind regarding backups: there are two kinds of hard drives: those that have failed, and those that will.