Unlimited. You can create and store **as many Xbox Live profiles as the hard drive / memory unit capacity allows** (each profile ~4–8 MB + save data). There is no system-enforced limit.
Guide → Profile → Create New Profile → Sign In with Microsoft Account (or Create New) → Choose "Save on This Xbox" or Cloud.
Note: Xbox 360 Marketplace closed July 2024. New accounts can still sign in, download *previously purchased* content, and play online, but cannot buy new 360 content.