I just found two open Dept of Navy positions on USA Jobs. One is GS Grade 05/07 the other is 09/13. Is it appropriate to apply for both?

If you are qualified. I would apply for the higher grade one and then, you can always accept a lower grade to start if that is within your economic needs.

Yes, apply for both

as long as you qualify for both positions there in no reason not to apply for both.

Depending on your experience, and if you would accept the lower position, Yes. But consider that any company or corporation wants to get the best talent at the lowest cost.

Are you qualified to do both?

I wouldn't, USAJOBS are Bullshit. I tried multiple times, got called a couple of times and never called back. Yeah, I know I gotta stop being so damn honest.