Before writing off renting a server, there are some serious advantages you may want to consider...
You get a dedicated, high speed line, no parts to replace, reliable uptime, support, and probably a lot more I've left out... Hosting your own rig, I assume you mean to run it at home? There's a good chance you wont have the internet connection you'd want for this type of thing, plus power redundancy, and a lot of other intangibles.
Anyway, when it comes to the hardware, you'll want to figure out how much RAM you need for the project you're doing, and Xeon CPU's are fairly common, which knocks down the price on eBay. In my opinion, you might want to consider something like an HP workstation, for example, a z620. It is a desktop style with server class parts. But really, do consider a VPS first, a dedicated server second. It might meet your requirements and save you a lot of extra hassle.