You're going to have to buy a used computer and put upgrades in to get that low of a price. Pentium 4 @ 3.2 GHz and a 9400 GT graphics card, along with 2 GB of RAM is enough to play Left 4 Dead 2 for example. How I know this is because I have upgraded an old Dell Optiplex GX280 into a budget gaming / business machine for when someone wants to play PC games and I'm using my high end system. The Optiplex started with a 3.2 GHz Pentium 4 with HT, 512 MB of RAM, and 64 MB integrated graphics. This PC is from 2004 and I bought it used off eBay for $75. I upgraded it to 2 GB DDR2 (this is when I learned that they had DDR2 machines in 2004
) and put in an Nvidia 9400 GT 1 GB graphics card. Pushed the PSU to the limit. Minimum listed PSU for this card is 300 W. That's exactly what this Dell has. After the upgrades, I installed and ran Left 4 Dead 2. Lag was minimal, but had to be on lower detail settings. And multitasking? Forget it. You're not going to want to play music with Media Player and play a game at the same time when you're running on a single core processor.
TL;DR: A game console is your best bet here. Even a used PC with upgrades will not get you that great of a gaming experience.