It may get me in trouble to mention, but will anyway.
For the ps2, one can use either the official network adapter and freehddboot+open ps2 loader on a FAT (and thus totally bypass the eol laser assembly-- even though doing so is a massive dmca violation-- so no, I wont tell you how or where to look, only that the tools exist, and what they do), or freemcboot + mx4sio (with microsdcard) + open ps2 loader for slim (which accomplishes the same thing above, and is ALSO a massive dmca violation, and no more details will be given)
The 360 will require a modchip, or custom firmware. No further info from me, but many emulators exist that were ported for such modded consoles.
A far more LEGIT solution, would be a MIST or MISTer-- which I feel I might actually be allowed to talk about in terms meatier than 'it exists, and does this thing'.
MISTer uses a small fpga core to dynamically simulate (not emulate!) Classic consoles, and is rather space efficient. It's harder to find, as it uses a difficult to obtain special fpga board meant for a very different purpose, but it is much more capable than its older sibling, the MIST.
The MIST is less capable (smaller --or rather, the the FPFA has less space to hold a core image, and thus can only do smaller cores, it's not 'slower', it's 'smaller'.) but readily available.
https://amigastore.eu/en/358-mist-midi-fpga-c … idi-add-on.html
multiple cores exist that can be deployed on it:
https://github.com/mist-devel/mist-binaries?search=1