Try booting from a PartedMagic CD or flash drive and see if you can access your files through that. PartedMagic is a live Linux variation that comes with a hard drive diagnostic program--something that you may find useful. You download PartedMagic in the form of an .iso file. If you double-click an .iso file, it will likely launch a program that will walk you through the process of turning the contents of that file into a CD (simply copying the file to a CD won't work). If such a program doesn't come up, download and install the ISO Recorder for Windows XP and Server 2003 or ISO Recorder for Vista and Windows 7. Then double-click the .iso file again.
See The Bootable Maintenance Flash Drive for instructions on putting PartedMagic on a flash drive.
If you can access the files while running PartedMagic, you can easily drag them to an external hard drive plugged into one of your USB ports.
And if you can't, try PartedMagic's drive diagnostic program, Disk Health.
If that doesn't help, you'll have to go on to the next step: Removing the hard drive from the computer and attaching it to another computer as a secondary drive (not the one you boot from). You can do this by either installing it as a second drive in a desktop PC, or by using a SATA-USB adapter (they cost about $20) to turn it into a temporary USB external drive. You'll then be able to plug it into one of the other computer's USB ports.
Once it's attached to the other computer, see if you can access the files. If you can, copy them to another drive.
Your last option is to send the drive to a data retrieval company. The two best-known are Kroll Ontrack and DriveSavers Data Recovery. Does that make them the best choices? Not necessarily. I know of no practical way to test these services, so I can't say that they're any better than their lesser-known and less-expensive competitors.
These services are very expensive. A successful retrieval (and there's no guarantee of success) will cost you hundreds or even thousands of dollars. But frankly, that's the cost of not backing up.