If the Hard Drive is still good, it should be possible to put it into another computer. One way to do this is to get a USB 2.0 to IDE / SATA cable. It looks a lot like this. (I got mine at Fry's, not amazon.com) This isn't a long-term solution, but allows for quick access. (My DVD burner mostly lives in a box, and I have the cable to connect it to whichever computer if I want it.)
If you have no tech experience, you might get a geekier friend for help.
Or, go to a computer repair shop and see about having the contents COPIED. (I remember someone who asked for a backup, and the tech shop was in a hurry and didn't do it right - then wiped his HD for a reinstall which failed.) You don't want that drive wiped. HD's are too cheap to justify it.
It sounds like your old computer may be old enough that replacing a MotherBoard or Power Supply wouldn't be justified (parts change rapidly and aren't often backwards compatible), unless you can re-use an used computer. Of course, if it's a fairly new computer, it may be worth the replacement.