I have notice that stackoverflow.com does not have file extensions on their pages. How would I do this with an aspx web site?
The URLs aren't actually pointing to files. They're using URL rewrite rules to convert the URL to a database query and feed the output back to a specified page (whose URL isn't displayed).
Here's a great article from Scott Guthrie:
http://weblogs.asp.net/scottgu/archive/2007/02/26/tip-trick-url-rewriting-with-asp-net.aspx
It covers off all scenarios: IIS 6, IIS7, using third party URL rewriting tools, etc.
Stack Overflow uses ASP.NET MVC which does clean URLs out of the box.
Basically what you need is something that takes the clean URL and then maps it to a standard .NET URL with your passing in extra 'directories' as parameters. e.g. rewriting /blog/post-no-one to /blog.aspx?id=post-no-one.
The new IIS has a rewrite plug in that will do this for you if you want to do it with traditional ASP.NET:
Or you can roll your own by overriding HttpModule and doing rewrites in there. Here is a complex example of that:
As a few people have mentioned, SO is using the URL routing engine included with MVC. They've actually made the code available via CodePlex, so you can use it inside of a webforms-based ASP.NET site.
Phil Haack has a nice summary of how to do this:
It includes samples, links, etc. that should be helpful in getting you started.