Your hard drive is corrupt
I have a 2011 21" iMac. A few days ago I kept getting an error message saying 'The disk' could not be read and giving me the option to eject or ignore the message. There was no disc in the drive and whenever I clicked eject the message would simply reappear. I used disk utility and it told me that the hard drive was corrupt and needed to be repaired, that I should reboot in recovery mode to use disk utility to repair it. However, the disk was always greyed out and couldn't even be verified in recovery mode, let alone repaired. Apple recommended, as I'd backed up, that I try and erase the disk - which also proved problematic - and do a clean install before importing a back up from Time Machine. I did this. I didn't even use a whole system from Time Machine, merely imported applications and photos and music etc. The machine seemed fine. I even used Disk Utility again to check the state of the hard drive and it said all was fine. Then, without having installed any new software or done anything, the message about the disk not being read with 'Ignore' or 'Eject' reappeared. I verified the hard drive again through Disk Utility and again it said the hard drive was corrupt and needed to be repaired. Again, in recovery mode I cannot repair the Mac HD as it's greyed out.
Is the hard drive simply faulty and needs replacing? If so, why did disk utility at one stage, probably only for a few hours!, give it a clean bill of health?