Macbook Air - White/Grey Screen with a folder and a question mark

Out of the blue , my macbook air stopped starting up and strted to show me a white/grey screen with nothing but a folder and a question mark . After searching on the internet for solutions and trying to reset my PRAM setting , it still wouldn't boot up . I opened up disk utility and going through internet recovery , I ran the first aid verification on my MAC OS X Base System and it said that it found no problems . I can't restore or erase by the way and i really wouldn't erase my files as I have very important files on my mac . Moving on , I tried to reinstall OS X , but apperently it needs a disk to install OS x on and since none appear , I tried using an USB . After creating a partition and erasing everything on my USB , not happy about that , it wasn't big enough for the OS . So now I am basically stuck in the mac white screen Limbo .

Any sort of help would be fantastic and please do keep in mind that this is a macbook air and I do not have a place to put DVDs into . All I have read on different forum pages didn't help so i decided that it would be a good idea to create my very own thread . I am not very sure about the Mountain Lion version ,but I am not very sure that it matters .