Unlike iOS, Apple's new tvOS does not provide support for persistent local storage. From Apple's App Programming Guide for tvOS:
There is no persistent local storage for apps on Apple TV. This means that every app developed for the new Apple TV must be able to store data in iCloud and retrieve it in a way that provides a great customer experience.
iCloud might be fine for iOS-only apps, but it doesn't work if you also need to support other platforms such as Android.
Fortunately, Apple apparently does provide a limited form of persistent storage on the new Apple TV. The Keychain can be used to store user credentials, which in theory can be used to authenticate the user to remote, non-iCloud services. See this discussion in the Apple developer forums:
Thanks to developer forum user TidBits for this very valuable piece of information!