HHD full and computer won't start. Help!!

My 500gb hard drive seems to be so full that the computer is unable to start. I turn the computer on and the Apple icon and losing bar appear and eventually it gives me this screen:

Please help I am totally buried in work and I cannot afford to lose some things I have on this hard drive. I've tried holding down the shift key to start in safe mode but it doesn't work. I can get to disk utility by holding command+R but I can't delete anything from there to make room on the hard drive.