XboX isnt possible. Currently, the only active Xbox emulator is still a ways off before being considered usable. It only boots several games and only a few of those are playable.
Gamecube is possible Via Dolphin.
PS2 is possible, but the only Proven emulator for PS2 is PCSX2 which is x86 only and not open-source, meaning whichever platform you use has to either be x86 or have an internal PS2 emulator.
Seems to me that your best bet is likely an x86 mini-PC build using BigBox. That said, Storage WILL be an issue. PSX alone is 500+GB for just the complete US RomSet. PS2 has significantly more games and many of them on DVDs rather than CDs (4.5gb vs 700mb), Gamecube discs are mini-dvds so roughly 3x larger than your average CD.
If you want to play the entire GC and PS2 library without issue.... I'd suggest a gen 2 Ryzen APU. Alternatively, you can pick up a 4th gen or better i3 and a gtx 750 or better gpu. Both will probably need 8gb RAM because Win 10 and large BigBox collections can be resource hungry. But by far, your largest issue will be Storage. 2TB minimum for highly curated sets for those two consoles. Also, a bonus, Wii should work reasonably well on any system that can emulate Gamecube.
You can easily consolize a PC to do what you want, but if you were hoping for something as small as a Single Board PC like the Raspberry Pi for this project, we are likely still a several years away.