Anything more than two cores is a waste on current console emulators. There are small mobile chipset motherboards that take the Intel mobile processors from the previous architecture. They can be had for around $400 and will take 4-8 GB of ram and use a fairly modern chipset like the PM45. I think Commell still makes them with a PCIe X 16 slot. You could get a passively cooled single slot graphics card for modern emulators like Dolphin and the Model 3 emulator.
I think this MB and cards/cables, etc will fit in a very small case, probably around the size of a Mini-ITX case.
Then, get a small SSD for the OS and apps. Put the roms on a high-speed network share for lots of room for full sets (like MAME)
Then, put a good front-end like Hyperspin and you're good to go.