How to copy files from an unbootable Mac?
Solution 1:
Boot up in Recovery Mode (Command + R) add attach an external drive via USB. Then open the Terminal to copy files using the command line. All disks are available under /Volumes.
Solution 2:
You might try using Target Disk Mode - hold down T at startup and then connect the MacBook Pro to another Mac (or a PC that can read Mac drives) via FireWire or Thunderbolt.
Or try holding down the Shift key to boot into Safe Boot mode.
If neither of those work, try booting from your MacBook Pro's system restore DVD, use Disk Utility and try repairing the disk, or try a 3rd party utility such as the venerable DiskWarrior.
Solution 3:
I recommend using a Linux Live disk, specifically...
Linux Mint
... as is comes with nautilus gksu
preinstalled. (This allows you to easily open folders as admin.)
You simply boot from the Linux Live disk, and use it to access your files on the Macintosh HD as administrator which allows you to copy them to an attached external hard drive.
Here's a short video tutorial on youtube. Note, that this tutorial uses an older version of Linux Mint.