I think I'm to the point where I have to erase my hard drive. . I don't know how to backup the system.
I have a external hard drive. How do I reformat my hard drive?
Have you tried booting into your Recovery partition (holding down the command and R keys whilst booting) and verifying your internal drive and repairing if necessary?
While in Recovery mode you can also reinstall OS X without erasing your data on the drive.
Try it and see what happens.
Retrieving data ...