You don't need that for flash, either.
What you need for flash is to have the video in a format that Flash can play. For instance MP4/h.264/AAC.
What you need for <video> is to have the video in a format that the browser can play. Different browsers support different formats. In practice this means you need two versions of your video: MP4/h.264/AAC (baseline and low complexity profiles, respectively, if you want it to play on smartphones) and WebM.
If your server supports HTTP Range requests then it's possible to seek in the video without having to download the whole thing first, so it's good if your server supports that. You will also need to have the correct MIME types for the videos for them to play in some browsers (video/mp4 and video/webm). But other than that, a standard HTTP server works fine.