Your best bet in my opinion is a big fat Xbox 1. They go for relatively cheap now, and despite what one guy said YOU DON'T NEED A MODCHIP.
You can use a soft mod by using a memory card and a original copy of 007 Agent Under Fire, Mechassualt, and Splinter Cell. You will need the ORIGINAL (not a greatest hits copy). You find a ton of information on it and how to do from the site Xbox-Scene.
There is an emulator for pretty much every system up to PlayStation 1. The compatibility on Xbox is damn near flawless, with the most problems coming from PlayStation and the N64 emulators.
As far as I remember, the SNES (Runs all except Doom, but that's just one emulator, the other for Xbox does run it) and GBA emulator run every game, but you probably will run into problems with the games every one else does, aka Pokemon GBA games.
You really can't go wrong with homebrew for Xbox 1. It's probably the BEST homebrew solution.
As far as portable devices, the PSP is really about the only choice. It can run PlayStation games if you convert them over. It does have problems with GBA though, as the last I remember no emulator has any RTC support or anything.