Step 1
The first step in dealing with a startup failure is to secure the data. If you want to preserve the contents of the startup drive, and you don't already have at least one current backup, you must try to back up now, before you do anything else. It may or may not be possible. If you don't care about the data that has changed since the last backup, you can skip this step.
There are several ways to back up a Mac that is unable to start. You need an external hard drive to hold the backup data.
a. Start up from the Recovery partition, or from a local Time Machine backup volume (option key at startup.) When the OS X Utilities screen appears, launch Disk Utility and follow the instructions in this support article. You'll be "restoring" the startup volume to the backup volume.
If you use FileVault 2, then you must first unlock the startup volume. Select its icon ("Macintosh HD," unless you gave it a different name.) It will be nested below another disk icon. Select Unlock from the File menu. Enter your login password when prompted.
b. If Step 1a fails because of disk errors, and no other Mac is available, then you may be able to salvage some of your files by copying them in the Finder. If you already have an external drive with OS X installed, start up from it. Otherwise, if you have Internet access, follow the instructions on this page to prepare the external drive and install OS X on it. You'll use the Recovery installer, rather than downloading it from the App Store.
c. If you have access to a working Mac, and both it and the non-working Mac have FireWire or Thunderbolt ports, start the non-working Mac in target disk mode. Use the working Mac to copy the data to another drive. This technique won't work with USB, Ethernet, Wi-Fi, or Bluetooth.
d. If the internal drive of the non-working Mac is user-replaceable, remove it and mount it in an external enclosure or drive dock. Use another Mac to copy the data.
Step 2
You might be able to start up in safe mode even though you can't start up normally. Otherwise, start up from an external drive, or else use the technique in Step 1b, 1c, or 1d to mount the internal drive and delete some files. According to Apple documentation, you need at least 9 GB of available space on the startup volume (as shown in the Finder Info window) for normal operation—not the mythical 10%, 15%, or any other percentage. You also need enough space left over to allow for growth of the data. There is little or no performance advantage to having more available space than the minimum Apple recommends. Available storage space that you'll never use is wasted space.
