Why not create another project, a class library, and put it in there. That’s by far the best way to keep those classes separate from the rest of your website.
To answer your specific question about the App_Code folder, it’s available in WebSite projects as an Asp.Net folder you can choose from. But from a web application project, you can just create a new folder and name it App_Code and it will work just fine. This is as of Visual Studio 2010 RC.