I’ve revised (but not changed) this question for brevity; see edits for previous version(s)
I have an ASP.net website (non-application, non-MVC) with the following (proposed) file system hierarchy:

What configuration changes (if any) can be made to implement this scheme? Is this possible? I’ve received the error Cannot use a leading .. to exit above the top when having targeted /Site as the application. I’m wondering if there’s a way around this.
As I see it, given the site runs in C:\www\SiteRoot\ at http://example.com/:
PATH URL
~ C:\www\SiteRoot\ none
/ C:\www\SiteRoot\Site\ http://example.com/
~/Site C:\www\SiteRoot\Site\ http://example.com/
/About C:\www\SiteRoot\Site\About http://example.com/About/
Therefore, MasterPageFile="~/Site/About.master" will still resolve correctly,
as will <img src="/About/ImageInAboutFolder.jpg" />. Hopefully this sort of configuration is possible.
If I understand your question correctly, no, you can’t.