Can You Save Xbox 360 Games Without a Memory Unit?

No. Every Xbox 360 requires a storage device for game saves:

• Internal HDD (20/60/120/250/320/500 GB) – original 360 models.
• Internal 4 GB flash (Xbox 360 S/E base models) – counts as a memory unit.
• USB flash drive (formatted via System Settings, up to 32 GB per drive, max 2).
• Cloud saves (Xbox Live Gold required) – still need local cache for initial write.

Without any of the above, the console will prompt "No storage device found" and refuse to save. Even disc-based games cannot write to the disc.