Is Xbox 360 emulation possible on Steam Deck?
Yes, but with real caveats. The Xbox 360 was a powerful console for its time, and emulating it accurately takes far more processing power than emulating something like a PS2. The main emulator for the job is Xenia, which has a version built for handhelds. Results are inconsistent: some games run close to full speed, others struggle to stay playable, and a meaningful share aren't compatible at all yet.
What to expect on Steam Deck's hardware
Steam Deck's APU wasn't designed with Xbox 360-level emulation as a target, so this sits closer to the edge of what the hardware can do rather than a smooth, guaranteed experience. Expect to spend time testing individual games and tweaking settings rather than a one-size-fits-all setup. If Xbox 360 emulation specifically is your main goal, a more powerful system — like a Steam Machine with its desktop-class AMD hardware — will give you noticeably more headroom.
A note on game files
We're not going to walk through sourcing game files here, and for good reason: downloading copies of games you don't own is copyright infringement in most places, regardless of whether the console itself is old. The legally clean path is dumping your own discs from games you already own — there are established, documented methods for this that any Xenia or emulation-focused community can point you to. If you don't own the game, emulation doesn't change that.
Getting started the right way
If you're set on trying it: install Xenia through your usual emulation frontend, start with a game you personally own and have dumped yourself, and check community compatibility lists before investing time in troubleshooting a title that's known not to run well yet. Treat it as an experimental feature of your setup, not a core one.