Mac shuts down automatically on gray apple screen when starting up

My mac was having the issue of the curser being a color wheel and just spinning on startup. I have cleared applications list, verify and repaired permissions but nothing has worked to keep my mac from locking up and spinning. Finally I restarted pressing cmd and "R" to get to disc utility to find out I need to reformat. But now when starting up gets to the gray screen with the apple logo and shows the load bar underneith and can not get to the desktop. Once it loads my mac just shuts down. This happens over and over if I continue to restart. I have accessed the disc utility from startup and tried to verify and repair but it says that I need to reformat because of a hard drive error but I need to backup my mac before doing that since I have a LOT of sensitive information on my computer. The issue is, I can not boot to desktop regularly or in safe mode to backup my mac. This is my work computer so I am unable to do anything at work until this gets fixed which means I am loosing money. Please help me boot to my desktop so I can get this backed up and reformated asap.