What to do if your Mac has started from Recovery HD
If you have started from Recovery HD, then with luck your Mac will see the offending drive. At this point, run the Disk Utility from the Recovery HD and see if that helps. If you have any disk management tools, like TechTool Pro, Disk Warrior or Drive Genius, this is a good time to try them out. They all claim to be more effective than Disk Utility, but how well they do is very dependent on whatever is at the root of the problem. However, if you attempt to use these tools on an SSD, you'll most likely find they aren't quite so helpful. The way SSDs work is very different from HDDs that most traditional tools struggle to help at all.
One thing to bear in mind is that if you do manage to boot from Recovery HD, you'll need another drive to recover files to. If any of the disk utilities manage to fix your drive enough to let you boot from it again, be very careful how you use it. You should plan to get an external drive plugged in as soon as you can, so that you can secure your data there. If you have lost any data, it is very important to avoid using the drive, as anything that writes files to the drive, may be writing over your lost files. Which would render them unrecoverable.
So if you manage to resurrect your sickly drive, quit all your software, including email, to avoid overwriting any list files. Once you have an external drive, you can then use that as your boot drive, and recover any lost files to there.
What to do if the Recovery HD is missing on Mac
If you are using an old Mac running an OS older than Lion, then there will be no Recovery HD option. Lion has been around since 2011, so anything newer than that should have the Recovery HD partition available if the drive is showing any signs of life. If you have an older Mac, you will need to boot from an external drive either optical or HDD, that has OS X on it.
If your machine is running Lion or newer, and the Recovery HD isn't showing, then it does suggest the drive problems are terminal. But you may as well try to boot from an external drive running OS X.
How to boot from another drive on Mac
If you are able to get either a DVD drive or an external drive with OS X already installed, you'll be able to boot from another drive.
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