My brother's MacBook Pro had a similar issue a few years ago. In his case the issue started when he attempted to update the OS to the then most recent version. The update failed with an error saying that the Hard Disk needed to be repaired and instructing him to restart in Recovery Mode and repair the disk. When he did this the "Repair Disk" option was grayed out as you said. From my searches then it seems that the drive entered a "write-locked" state that prevented it from mounting. I was able to recover his files and restore the computer to working order. You will need another computer that has enough free space to hold the files you want to save, and the appropriate cable to connect the misbehaving computer to the working computer.
This procedure requires booting in Recovery Mode and erasing the non-working computer's hard drive, reformatting it, and restoring the system.
1. Turn on the working computer.
2. Make sure the non-working computer is turned off, connect the two computers using the appropriate cable (FireWire or Thunderbolt)
3. Turn on the non-working computer holding down the "T" key to enter Target Disk Mode. (This worked in the case of my brother's computer, but I can't guarantee that this will work in your case, but if you could get it into Verbose mode I would think you should be able to get into Target Disk Mode.)
4. The non-working computer's hard drive icon should appear on the working computer's desktop. You may get a message that you cannot alter the contents of the external drive (non-working computer), but you will be able to transfer files from the non-working machine to the working one.
NOTE: You could connect a dedicated external drive to the working computer and transfer files from the non-working computer to the external drive using the working computer as an intermediary of sorts.
5. Transfer the files you wish to save. We will be erasing the non-working computer's drive and restoring the system so be certain you have what you need transferred before you get to that step.
6. Once you have saved all of the files you need to save, you can eject the non-working computer's hard drive, and turn off the non-working computer.
7. If possible connect the computer to the internet, if not possible you can use wi-fi, but it may take longer.
8. Connect the non-working computer to the drive with your most recent backup.
8. Turn on the non-working computer holding down the "Command+R" keys.
9. Once the computer is in Recovery Mode choose the option to Restore From Backup
10. You will then be asked to choose which backup to use, and then click "Restore". Once this process is done you should be able to boot the system. You can then transfer the files you saved to the other computer or external hard drive back to the restored computer.
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