This is one of these ideas that is bound to upset most of your visitors - apart from the fact that it is not possible as a ‘true’ background.
You might be able to have the video play behind some objects but content that is displayed via plug-ins tends to ‘float’ to the top.
Also a video that is a decent enough quality to play full screen on some of the larger monitors would be a pretty hefty file size and probably a slow page load to match.
Maybe if you can give us an idea of what effect you want to create we can offer a better solution.
Have you seen this somewhere and you wish to emulate it? If so then post a link to the example.