How do you run a gnu screen within a gnu screen?

You just hit ctrl-a followed by an a to actually send the ctrl-a to the nested screen instance.

Yes, it is possible. You will need to redefine screen's escape character, e.g. set it to E when you call screen

screen -e^Eq

If you clear the $STY environment variable, running screen will start a nested session.

You then have to send a literal ^a to the nested session before using any of your bindings. You can send a literla ^a by pressing ^a followed by a (plain).

So to create a new window in the nested session you would press ^a, a, c.