Nope, it require a few megabytes or less for the app itself, and zero storage for the streaming. It's instant 1080p streaming with no wait, and no huge buffer, as long as you have at least 8-10Mb or so of bandwidth. If you have less, it uses a lower resolution, I think all the way down to 2Mb or less. It works across multiple devices, including rokus and others with zero storage capability.
The time between you deciding to watch a show, and that show starting to play is generally less than 5 seconds. Sometimes it's as long as 10 seconds (and that feels like an eternity nowadays - how spoiled we are..), most of the time it's about a second or two.