Hard drives are cheap. Pull yours and put it in an external case. Install a new one internally and load a clean system. Then access the bad drive externally. Use a file program to access the external disk and get your data files off of it, then reformat it.
You can also direct your computer to boot off a recovery partition, and use that to format and load a clean system on a new external hard drive. Once done, boot using the external hard drive and get your data files as you mentioned. Then you can either reformat the old drive, and do a swap of drives, or clone from the external to the old drive.
I just bought some refurbished hard drives for under $20 each to use for purposes like this. As you said, hard drives are cheap. I regularly clone my internal drives so I can boot off the external clone as needed. Makes for a clean guaranteed backup if necessary.