Many times when systems have troubles that may take an extensive amount of troubleshooting to sort out, the easiest solution is to perform a system reinstallation. Before doing this, though, it is always advised to have a backup of your system (or your data at the very least); however, in instances where you cannot boot your system, this may seem impossible to do. Despite this, if you have a system that cannot boot but you need to back up, there are some options available to back up the drive before wiping it clean.
The first thing you will need is a working storage device such as an external USB drive that has the capacity to back up the contents of your faulty drive. This is important to have ready from the beginning because sometimes faulty drives can still be unstable even if you can read them when you've accessed them in alternate ways, and being able to quickly transfer what you can off of them can be crucial in saving your data.
When you have a secondary storage drive available, the following options should allow you to access the faulty boot drive and get your data.
Target Disk Mode
If your system has a FireWire port, then you can boot it into Target Disk mode, which will turn your entire computer into a an external hard drive. Boot while holding the "T" key down, and when the FireWire symbol appears on the screen, attach it to another Mac with a FireWire cable.
Attach your recovery drive to the second Mac (or use the second Mac's hard disk) and copy the contents of the faulty boot drive to the new Mac.
Another similar option is to purchase an external controller or drive interface cable that will allow you to connect the bare drive to a system via a USB cable. When troubleshooting or managing hard drives, these adapter cables such as the NewerTech Universal Drive Adapter can be priceless.
You will likely need to use a second Mac to copy the drive's contents using an adapter cable, but another option is to replace your current Mac's drive, install a fresh copy of OS X, and then use the healthy installation to attempt recovery of your data from the faulty drive.
Secondary or external hard drive
If you have a spare external hard drive, then you can use that as an intermediary on which to install OS X and boot the system. Then you can use that installation to manage the data on your normal boot drive. If your external drive is big enough, then one option is to install the OS to this drive and then try using the OS X migration assistant to transfer all of your data, applications, and settings to the new drive. After your data is migrated, you can use a cloning utility (or even the same process with Migration Assistant) to transfer the data back to the main boot drive at a later point.
Regardless of the method you use to access and back up the data on the faulty boot drive, the drive must be readable. Therefore, if your drive is only suffering from a software configuration glitch that prevents it from booting then you are in a relatively good situation; however, if the drive will not mount or has other filesystem corruption, then even if you use the above options you may still have problems accessing your data. In these situations, using robust disk utility software may be your only hope. Tools like Drive Genius or Disk Tools Pro are great at rebuilding disk volumes and filesystem structures, but the most reputable option for rebuilding volumes is DiskWarrior. Apple's Disk Utility is a good starter, tool, but many times is not enough to tackle major directory corruption, and it will not repair faulty partitions.
Run these tools' repair routines on the faulty drive to see if you can get the drive working again, and if so then immediately back up your important data off it. While the drive may be working again, the fix may only be temporary, especially if the drive is experiencing hardware problems.