Caution: You will be directly editing the raw data on your disk. It's entirely possible to make things worse this way! If you don't know what you're doing and there's valuable data on the disk, it may be time to consider a professional data recovery service. However, I have done this more than once (due to boneheaded moves like installing GRUB on the wrong partition) and it's always worked for me.
To start with, you need a good disk sector editor. The one I'll be using in this example is PhysTechSoft Disk Editor. Beg, borrow, or steal a working system, download the files, and run MAKEBOOT.BAT to make a standalone boot floppy.
Boot the floppy. Now click the Partitions radio button and click your damaged partition. You'll find yourself looking at the first sector in the partition -- the damaged boot sector. Note the "Absolute Sector" number above it.
Since Microsoft has thoughtfully provided a backup copy of the boot sector at sector 6 of the partition, and we're looking at sector 0, add six to that number. Now click Search, Goto, and type in the new number. For example, if the first sector of the partition is sector 63 (a common value), you'd type in 69.
You should see the familiar sight of a DOS boot sector (If it's not a familiar sight to you, check out the example fragment in my other boot sector instructions.) Select the entire sector, then click Edit, Copy. Now click Search, Goto and type in the number of the first sector of the partition (63, in our example.) Click Edit, Paste. Now click Window, Save sector to save your work.
That's it. Exit and reboot, and you should be in good shape.