Macbook Pro froze, I used command option power button to reset, turned computer back on, and now it flashes a file folder with a question mark?

So I shut it down, and I googled what to do on my phone, and it told me to hold option while I hit the power button.

All I can get it to do is open the internet recovery window, where it gives me the option to reinstall, restore from backup, get help online, and run disk utility

I never backed it up (stupid stupid) and when I open disk utility, it can't show any information about my hard drive and it won't mount, and the verify and repair buttons are grayed out.

I would prefer to fix this and save all my files, but those of that seem slim so I was wondering if it was possible to back up as many of my files as I can, then reinstall.. I read somewhere about installing to another harddrive, booting from that, and then backing up my files, then reinstalling my harddrive.. Any opinions or solutions? Thank you!