Backup files from recovery utility
There are a few methods to accomplish this:


If you have another Mac accessible, boot the dying Mac into Target Disk Mode. Plug it into the other Mac via Thunderbolt or Firewire. You should see your dying Mac show up as an external hard drive. Drag and drop what is needed.
Get an external hard drive at least the size of your Mac's internal drive. Plug it in and boot into Recovery Mode. Open Disk Utility. You should see the external drive and boot volume. Use Disk Utility to Restore one volume to the other. This will destroy any existing data on the external hard drive.
If you can't reformat an external hard drive but have a bit of space available to move the select files you need, you can boot into Recovery Mode and use Terminal to move the files over that you need. Use the cp UNIX command to do this. For example, to move all of your Documents to an external hard drive named "External" you would use the following command:  

cp -r "/Volumes/Macintosh HD/Users/joe/Documents/" "/Volumes/External/"