Mac OS X Lion on Macbook Air 2012 model froze

The laptop froze on me, and then I restarted and got the question mark folder. Afterwards, I held the option key and made it to Internet recovery.

Now, I am trying to "Reinstall Mac OS X" but when I go to the screen that says:

"Select the disk you want to install OS X"

It is blank (ie. no option is available).

Also, I tried going on Disk Utility and its really restricting my options:

1. The "Erase" options are grayed out (ie. I cannot select them).

2. "First Aid" keeps telling me that "The volume Mac OS X Base System appears to be OK"

But the "Repair Disk" option is grayed out.

3. I am unable to create an image (I keep getting an error -61), so I cannot do "Restore"..

What can I do? I need to ultimately restore my files...