Not sure why "33% overhead" is always mentioned, when that's complete nonsense. Yes, it does initially roughly add that amount, however there's a little thing called gzip (ever heard of it?). I've done tons of tests and the difference is typically negligible. In fact, sometimes the gzipped base64 string is actually smaller than the binary file. Check out this guy's tests. So please, can we stop spreading absolute fiction.
Base64 is a perfectly acceptable method of retrieving a video. In fact, it works amazing for a private messaging system. For instance, if you were using AWS S3, you could store the files privately so there is no URL.
However, the main disadvantage (imho) of using a gzipped base64 video is that you need to wait for the whole video to load, so pseudo-streaming is out of the question.