My Mac Air Will Not Boot

I can turn on my Mac Air, select my admin account, and log in, but the bar loads only about a third of the way and then the screen goes black.

I've tried different solutions like resetting the PRAM or the SMC, but nothing has worked. It won't let me enter safe mode or try using fsck. When I use Disk Utility on Recovery Mode, it tells me that the volume was found corrupted and needs to be repaired by clicking "Repair Disk," but it won't let me click it. When I try to reinstall the OS, it doesn't allow me to select a disk.

Please help--I'm at a total loss, especially since I don't have my files backed up.