The only thing that may work for you is this (although, I wouldn't get your hopes up):
Start your computer and hold the ⌘ + S keys to get into Single User mode.
Once you're in Single User mode, type in
unmount -uw / and press enter.
Now, type in
fsck -fy and press enter. It should go through several stages of checking and repairing the file system.
It will most likely give you an error stating that it cannot be
completed for some reason, but it's worth a shot.
Alternatively, if the issue is isolated to the file structure, you can try doing a reinstall of Mac OS X from the recovery partition. Doing just a reinstall should not erase any of your data.
However, you may want to try to back up your computer first. The only way that I can think of to do this on a MacBook Air 2011 is to use Thunderbolt target disk mode. This will require that you have access to another Thunderbolt equipped Mac as well as a Thunderbolt cable.
If you don't, I'd take a trip to the Genius Bar at your nearest Apple Store. They'll have the extra Mac and equipment necessary to do this. If they/you decide to do this while you wait at the Genius Bar, expect to be there for a couple hours just in case. If they/you decide to leave it with the Geniuses, there is a $99 data transfer fee that they'll most likely charge you.
The only other option is to:
Boot your Mac to the recovery partition
Open Disk Utility from the Utilities menu
Follow these steps to erase your SSD
Reinstall Lion on your MacBook Air