Hi and Welcome. My Name is ***** ***** I look forward to assisting you today with your question.
This indeed an issue with the hard drive - we're sure not whether it is hardware or software related. Your Mac is trying to load the OS X from the hard drive but it couldn't and that is why it is shutting down. Other components like processor, logic board, RAM etc are fine. It is just the hard drive. The usually happens if partition of the hard drive is corrupted, that is why you are unable to repair it. Since it is corrupted, you will have to erase (format) the hard drive and reinstall OS X.
You have four options in here. First is, buy Disk Warrior (powerful disk utility application) and try to repair your hard disk or recover your files with it. If it is unable to fix the hard drive, then it will at least help you to recover the files that are readable and save them to an external drive. Get it from here : http://www.alsoft.com/diskwarrior/
Disk Warrior is costly. If you don't want to recover any files, then you don't have to buy Disk Warrior. It is not worth it if you don't have any important files/applications.
Second is, if you have another Mac and a Thunderbolt/Firewire cable, then you can setup Target Disk Mode and this will allow you to mount the hard drive of this Mac to that like an external hard drive and you will be able to backup your files --> https://support.apple.com/kb/ph10725?locale=en_US
(After backing up, you can reinstall the OS X after erasing the drive)
Third is, you can erase the hard drive and try to install a fresh version of Mac OS X in it. This will erase everything from your hard disk unfortunately. Boot into the Mac OS X Utilities screen just like you did before. In there, select Disk Utility. In Disk Utility, select your hard disk and erase it. Then close Disk Utility and continue installation. This is the quickest solution if you don't mind losing the files in it.
Final option is, you can take the Mac to the Apple Store. They will recover the files using their tools like Disk Warrior or Target Disk Mode and then reinstall Mac OS X for you. I'm sorry, but you have already tried all troubleshooting steps so far and these are your only options.
Please let me know what you think.