Actually, you can easily find an 8600 GT for under $100. It's definitely enough for GZDoom, and should work well enough with Half-life 2 and other games that are a similar age (or older), but it'll struggle with more recent games.
If you're interested in playing games like BioShock or Mass Effect, I'd recommend upping your budget a bit and getting something like the Radeon 4850 (which can be found for under $200). Don't pin your hopes up on running Crysis well even with that card, though

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