Downloading Games on Xbox 360 Arcade: Service Discontinued (2017) — What Still Works
You cannot download new games on Xbox 360 Arcade. The Xbox Live Marketplace for Xbox 360 closed July 2017. The "Arcade" model (no internal HDD) also cannot install disc games to storage.
What Still Works
Previously Purchased Games: Redownload from "Download History" if licenses intact.
Physical Discs: Play any Xbox 360 disc (backward compatible titles work on Xbox One/Series X|S).
Backward Compatibility: Hundreds of 360 games play on Xbox One/Series X|S (digital or disc).
Options Now
Play existing library on 360.
Buy physical 360 games (used market).
Upgrade to Xbox Series X|S for massive library (Game Pass, backward compat, new releases).