You don't need to physically remove the hard disk to back it up, you simply need to be in a working operating system and have an alternative storage device (like a USB key, or an external hard drive, or even the internet!).
"But Windows isnt working" you say.
A Linux LiveCD I say.
You need 1 working computer, a reasonable broadband connection, 1 blank CD and around 1 hour to make one. I'd probably suggest Knoppix for your purpose.