prohibitory symbol at startup

So, my MacBook (13 inch, late 2012) crashed in the middle of doing some homework, now upon startup its showing the prohibitory symbol. I do have windows installed as well via bootcamp, but it cannot start up either. I have tried repairing the driver and repairing permissions. I also attempted to reinstall OS X Lion through Recovery but I cannot as I am currently running OS X Maverick and it refuses to let me install an older OS.

I would prefer any solutions that don't involve completely wiping the disk as I do not have a recent backup of my hard drive with all of this semesters work on it.
I do have a external hard drive that is big enough to hold all data currently on my mac but disk utility will not let me back up onto it claiming it is the wrong format, even after completely wiping the drive and reformatting to every option provided by disk utility.

Any help is greatly appreciated.