The title of the question says it all. Is it possible? If so, please explain how to do it.
It may be possible with the use of WINE (or something similar, Mono, thats it.) but you will require all the .NET framework so it may be a bit much to setup
You can run most ASP.Net applications on Linux using Mono:
http://www.mono-project.com/ASP.NET
Mono is an open source implementation of (most of) .Net
I wouldn't try using Mono. You could use VMware or something but I would try go a different route. You can run a very basic ASP.NET site on Mono; but I mean very basic. If your site is non-trivial, I wouldn't recommend this approach.
It's possible to run ASP.NET on Linux. You should take a look at the Mono project, Mono has an implementation of ASP.NET and ASP.NET AJAX.
You need to use mono and XPS (which is the server that runs the ASP.NET framework on mono). There is a standalone version of XPS and one that works through Apache. That is the best way to serve the files on Linux.
As other posters have noted, you can use Mono and Apache with mod_mono. I have no idea whether it's robust and feature-rich enough for your needs. (Mono's current implementation of WCF is only skeletal, for example.) So whether it's advisable depends on what you plan on doing. If other people's money is riding on your applications being robust, it's safest to just plonk some money down on a windows server and going full Microsoft.
Update: in the comments, you suggest that price is a concern. In the long run, what you pay for hosting is tiny compared to the costs involved in using inferior software. I'd recommend either using Windows hosting, or using a different development environment altogether if you have a choice. LAMP, Java, Ruby on Rails--each of these are perfectly fine choices, and I'm sure there are many others. But if you are constrained to using .NET for some reason, Microsoft is really the only choice right now for enterprise-quality ASP.NET.