I made a frontend for Gmail prototype called “make-your-own-gmail”, that can be skinned as you see fit.
I hope this might help you wrap your minds around single page apps, and also empower you to believe that you can improve email even if you aren’t employed by Google or a major email provider.
If you’d like to host this at your own domain you’ll need to obtain a Google Developer’s key from the Developer’s Console.
I’d also highly recommend that you host it over HTTPS.
NB: email composition “kind of” works–there is some road to cover if you want to reproduce all of the features available to users at mail.google.com.