Metal Gear Solid V: Ground Zeroes requires a hard drive on the Xbox 360. This has been stated by Konami.
A warning on the box makes clear to users that they need a hard drive, and doesn’t support installing to a flash drive. Because of this, certain Xbox 360 systems (the Arcade and Core models) cannot play the game.
From Konami’s support page:
On the Xbox 360, in order to help you through your missions, a hard drive with up to 1.7GB free is required for the initial installation. Additional requirements are 192KB to save your game, and up to 256MB of free hard drive space for system updates and additional storage for some game features. Storage requirements are subject to change.
Source: Konami, Gamespot