While the earlier models had a small amount of storage, the current Apple TV (3rd generation) has no onboard storage for movies, TV shows, games and apps. It has a small amount of flash memory for buffering, but the vast majority of your content streams from other sources like Netflix, HBO and Hulu servers, your computer, your iPad, iPhone or iCloud, etc.
This will almost certainly change with the 4th generation Apple TV. Apple is expected to launch an Apple TV dedicated App Store for the Apple TV 4, plus a software development kit for third party app developers. The new Apple TV is also expected to be the hub for HomeKit and all of its apps, so built in storage is likely to return with the next Apple TV.