The documentation for this new software from Dragon Forged clearly states that running it can further damage your failing drive, but that's not such a bad price to pay if it can save your data. It requires a hard disk that can still mount, and features several means of rescuing your flailing files, including transferring single files (less likely to fail than batches), minimal scanning to reduce failures and error correction with no time limit on transfers. It can also target specific directories or even file types.
Thankfully, I'm unable to test the application right now. But if you're in a situation where this might (now, or eventually) be a lifesaver, Dragon Forged is offering TUAW readers 25% off the $20 pricetag until June 1st. Just enter coupon code "BHBB0008F" when you purchase. More information, documentation and a trial download are all available on the product page.