can you back up once your phone is in recovery mode?

My phone keeps randomly powering off. Now, it won't even turn back on when pressing the wake and home button.

I was instructed to back up my phone on itunes, before i restored, but my phone is now in recovery mode and "back up" isn't even an option in itunes, like the tech instructed. Is it even possible to back up when in recovery mode or is all of my stuff lost for good?