2015 iMac Will Power On, Won't Boot& Backing Up In Recovery Mode
My iMac (purchased late April 2015) will not reboot. Had computer on until I powered it off this morning, and about 10 minutes later while powering it back on (after updating), it would stay at the black loading screen (like when a new update is trying to load on the system) until about a 1/3 of the way, then shut off. Before shutting off, it asked me if I wanted to reload all the programs I had open, because the computer shut off due to an error (which is untrue, I powered it off from the menu), so I clicked cancel so it wouldn't reopen my opened programs. Tried restarting, no good. Tried SMC restart, still nothing. Did a PRAM reset and still no luck. Eventually ran Recovery Mode and Disk Utility; says my disk is no good (access points and such are just 1 number off from where they need to be). Can't run Safe mode to back up items because it just shuts off again while trying to boot, and can't reinstall Mavericks; it says my disk isn't good.
If there is no solution to this issue and I have to restore my computer, is there a way to back up my photos/projects/etc. in Rocovery Mode? Have some freelancing jobs in the middle of working, and my "backup to Time Machine" day is the first of each month; don't want to lose all the work. :/
System update should be the latest, because that's when the issues started occuring.
iMac (27-inch, Late 2013), OS X Yosemite (10.10.4), Purchased Late April 2015