If your computer does not have a hardware problem (drive, display, board,etc.), you can start up your computer with an external drive or disk that has a system on it. For older computers the system disk came with insallation disks.
If you are using OS10.9 (Mavericks), when you intsalled OS 10.9 the intsaller creates a "Recovery Drive" on your computer.
In either case, when you start your computer press the option key. You'll get a choice of drives to start up from, choose the external (or disk) or recovery drive.
If your issue is not caused by a hardware issue, your computer will start up and you will be able to access your user folder or any other folder and copy the files to an external drive.
I went to the Apple Store because I wasn't sure if I had a hardware or software issue. Fortunately for me, the computers tested OK and because everything was backed up, the "Genius" wiped the drive and installed a "fresh copy" of OS10.9.
Hint: Installing an OS over the internet at home takes hours. At the Apple Store wth their high speed servers, it took 10 minutes.
Good Luck!