Does Fallout 3 (Xbox 360) Have a Personal Vault/Stash?

No. Fallout 3 has no personal vault or player stash separate from your inventory. That feature debuted in Fallout 4 with the Workshop/settlement system.

Storage Options in Fallout 3

• Any Container: Footlockers, fridges, filing cabinets, ammo boxes—safe, non-respawning in owned homes (Megaton house, Tenpenny Tower suite).
• Companions: Trade with followers (Dogmeat, Fawkes, Charon, etc.)—they act as portable storage.
• Workbenches (DLC): The Pitt, Broken Steel add crafting benches with linked storage.

Fallout 4 Added the Vault System

The Workshop / Settlement storage (shared across linked settlements) debuted in Fallout 4. Not backported to Fallout 3.

Use your Megaton/Tenpenny containers—they're infinite and safe.