Start up in Single User mode
You can do a disk repair attempt by starting up in Single User mode (holding down Command-S at startup) and running Unix's fsck command. This method should almost never be necessary. However, if you find yourself with no other option, an Apple support article details exactly what to do.
You've run Repair Disk. Now what?
You've finally found at least one way to attempt a disk repair with Disk Utility's First Aid or its equivalent. Congratulations. Now what? That depends on the outcome of your attempt:
Your disk is OK: If First Aid reports "the volume appears to be OK," it's time to look elsewhere for the cause of your problem. Ultimately, in a worst-case scenario, a fix could require reformatting your drive, reinstalling a fresh copy of OS X, and restoring your data from a backup. For details on how to do this, see "Should you do a 'clean install' of Lion?" The advice still applies for Mountain Lion.
Your disk has a problem but First Aid repairs it: If First Aid reports a problem and is able to repair it, that's the likely end of the story. Conventional wisdom says to select Repair Disk a second time before quitting Disk Utility, just to be certain that no further repairs are needed. After that, reboot from the repaired drive and hope that all is fine now.
Your disk has a problem that First Aid cannot repair: If First Aid finds a problem but cannot repair it, you can try a third-party repair utility, such as Alsoft's $100 DiskWarrior, which is even compatible with Apple's new Fusion drive. Otherwise, reformatting the drive may help. It's worth a try.
Software utilities and reformatting cannot fix a physical problem with the drive. If your drive is making unusual clicking noises, it's almost certain you have a hardware problem. Assuming you've backed up your data, and given how inexpensive drives are these days, I would replace a drive before wasting too much time trying to resurrect it. If you can't replace the drive yourself (which is likely with recent Mac models, almost all of which Apple has made difficult to pry open), it's time for a trip to an Apple Store or Apple-authorized service provider.
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