I'm not suggesting full videos, theres already sites like youtube/vimeo for that and it would require probably require drastic changes to the CDN and storage.
Just as an alternative to animated gifs.
* Limited file sizes/lengths to sizes similar to animated gifs.
* Loop like gifs.
* No sound.
* No controls.
* Could even support a fallback to animated gifs when there is no support.
* Save file sizes/bandwidth
* Give better quality.
* Encourage other services to support the standard (forums and so on).
Compressing video into a sequence of pictures like we see with animated gifs is a silly hack.