Unlike the original Xbox, the 360 doesn't require a hard drive. Microsoft has sold millions of systems sans storage, and millions more with tiny 4GB drives. Most recently, the company started selling the 4GB system for only $99--with a required subscription fee. Unfortunately, anyone owning these consoles will have to upgrade if they want to jump into Halo 4's multiplayer mode.
Microsoft has confirmed that you'll need 8GB of space to jump into 343's online offering. From the official Xbox website:
At a minimum, you'll need at least an 8GB USB Flash Drive or an Xbox 360 Hard Drive to play Multiplayer mode in Halo 4. For an optimal experience playing with an Xbox 360 Hard Drive is highly recommended. For Xbox 360 4GB, Arcade and Core consoles, the Xbox 360 Hard Drive is sold separately.
Hard drives for the Xbox 360 retail for around $100 for 250GB of storage or more. The official Xbox 360-branded 8GB USB stick retails for $35--however, non-branded ones can go for significantly less.