The problem you have appears to be a damaged hard drive in a specific location. Back up programs will generally have a problem here as well. You can try, but there is no guarantee they will work either.
To do a full backup, you'll need another drive or partition you can put the back up on? Do you have one? If not, you'll need to buy one, and that would be a USB drive that you plug into a laptop USB port. Then you get Macrium Reflect (free) and use it to backup the C: drive to the USB drive.
Once done, you do an Install to the C: drive from your laptop install CD's. Choose the FORMAT option and if there is a choice of LONG, take that. Do not use the QUICK option. Once done, your drive will be smaller than before as bad sectors are removed. Then restore the backup made.
Again, depending what was in the bad sectors, something may not work anymore.
If the laptop is still under warranty, get the drive replaced. Many times when a sector goes bad, others will later. Not under warranty, consider replacing the drive.
There could be some utilities around that will recover damaged sectors, but I don't know of any that are free nor have any experience with any either.
Before doing anything check these links with other suggestions and answers :
Good luck. Personally, I'd be backing up needed data to CD/DVD, buying a new drive, installing Vista, programs, and then putting my data back if I couldn't make a good back-up.