Depending on what sort of machine it is, and your interest in tech work, there's a few things you can do. Fortunately, if it's just the machine refusing to turn on at all, there's a very high chance the data's mostly okay, and a decent chance the machine's repairable - depending, again, on model.

I don't know much about repairing/replacing power supplies, but it's something that is doable. If the motherboard's somehow fried, then the computer's a lost cause - but the data might not be. There exist external cases for hard drives, usually running 10-20 dollars. It's possible to open up a computer and remove the hard drive, sticking it in one of those cases, at which point it's effectively an external hard drive. I've got a drive from a dead laptop kicking around with ancient files stockpiled on it.

If you've got a basic tower - as opposed to an all-in-one machine - then a set of screwdrivers should be enough to let you remove the hard drive yourself. That said, having a tech-savvy buddy look at the machine to determine if it's repairable isn't a bad plan, first; whoever you talk to, ask about pulling the hard disk out and putting it in an external casing, if the computer's unsalvageable.

Hope that helps!