OS X Yosemite installation error: File system verify or repair failed

My Mac has been running fine until I decided to upgrade to OS X 10.10, Yosemite. When it is installing I get the message:

File system verify or repair failed.

I clicked the restart option to boot into my internal HD but tells me its unavailable. I then rebooted, held Command+R to enter disk utility in order to repair the disk, but it tells me it failed and should backup and restore the HD.

I then tried to create an image on my external drive but this also fails with the message:

Unable to create Macintosh HD.dmg. (Input/output error)

I am stuck in this vicious cycle of restarting and the installation failing. I cannot escape the installation process.

Is there a way to access my files and copy them onto my external HD, then erase and restore the internal HD?