3Ghz or go home.
Need to get 3.4+ if you want to get into the processor hogs like Blitz & Gauntlet Legends.
For Nvidia cards, x60 or better (460, 560, 660, etc) for the 3D emulators like Demul and Supermodel.
I was thinking about the $250 budget PC in my stand-up cab last night. Originally I thought I'd want to build a faster PC when the cab was done, but there aren't any games that I want to play that it can't. It might not play the new PC fighters with all the graphics settings maxed out, but it plays them at 1280x720 with everything on medium and it looks damn good to me. It plays Blitz & Gauntlet Legends. It chokes on the NAOMI shmups in Demul, but will play them on Makaron emulator. The specs are 3.4Ghz Athlon X3 (cheap $40 budget mobo unlocked the 4th core somehow), Nvidia 9800GT, 4GB RAM.
I've never tried a PS2 emulator. I've heard the 2D stuff isn't as demanding, but I know there is no way this would be enough for 3D PS2 emulation.
My driving cab has the same processor and video card and I've played Gamecube driving games on it using Dolphin emulator.
Some ran perfect, others had slowdown. I imagine the non-driving games are about the same.
I would use newer generation stuff if building today, but you could probably still put together a $300 system that runs all this stuff if you hold out for good deals.