A number of things could be causing this. It would be helpful to know a few things, such as how long you've had it, if you have made any changes, etc? My guess is that your processor may be fried, which is entirely possible considering the release date of the computer. Although it was only put in to production five years ago, this is about the point where some user's of said model will have processors die on them for various reasons, such as electron-migration. Just what electron migration is, you will have to google, it would take too long to explain here. Also make sure that the computer isn't filled with tons of dust. If there is too must accumulated dust, the cooling system won't be able to dissipate heat fast enough, if at all, and the system may overheat before it can get going, then shut off to avoid damage. You may also consider removing the heatsink fan and putting new thermal paste on it. I did this with my HP desktop recently and it dropped the temps an unbelievable amount. With the stock thermal crap it was always teetering on the edge of overheating while it was under load, and saying that the idle temps were toasty would be a gross understatement. Keep in mind, my desktop isn't even two years old yet, and the thermal paste was already going to crap. I'm not saying this is the case for you, but considering the (lack of) quality of stock thermal paste, it certainly wouldn't surprise me.
The last thing you need to look into is the possibility of your motherboard being fried. In my opinion, there is a good chance this is what you are dealing with, since you mention the case fans not turning on. I supposed this would also affect whether or not your hard drives power up as well, but I guess it's possible that it isn't in this case. If you think you have a bad motherboard, you can go on Newegg/Amazon and look around for a new one. I assume your CPU is based on the LGA478 or LGA775 chipset, but make sure you look up your CPU model on the manufacturer website just to double check. Also be sure to buy one that will fit inside your current case and that has the right kind of ports and connectors and whatever for your existing hardware.