Is it possible to run an XNA game without .NET or XNA installed?

No - The .NET Framework requires a fairly elaborate installation. You will not be able to run an XNA game without installing the appropriate framework.

You can't run your XNA game as-is without the appropriate install. However, there's an open-source library called SilverSprite whose goal is to "Run XNA games without code changes in Silverlight 3." Obviously there's some coding -- at the least, you need some new startup code -- but they give an intro on their project page and it looks like it would probably be pretty quick to get going.

Of course this would mean anyone who wants to play your game would need the Silverlight plug-in, but if that's available on your school's computers (and it might be), then you'd just need to open a Web browser and point it to your game.