The service you posted a link to appears to be a backup solution - not what you need at this point. That would help you if everything on the disk were previously backed up using that service.
If you have any linux/unix experience or have some patience and willingness to learn, I would suggest using a linux live disk and the command ddrescue. I believe it comes with the Opensuse live disk, and probably many other distributions. A live disk allows you to completely boot up an operating system from a CD, DVD or USB, without anything having to be installed on a hard drive (useful for troubleshooting and fixing things).
There is a difference between corrupt data and failing media (i.e., the hard drive). When the media is failing, you need to get as much of the data off as quickly as possible with as little stress to the device as possible. Once you have copied as much as you can to reliable media, you can then worry about dealing with corrupt data. In typical usage, the command ddrescue will copy one hard drive to another, skipping over sections that return read errors, then return to try harder on the problematic sections only after finishing copying the sections that are easily read. This can make a big difference in how much usable data is recovered, since you won't kill the hard drive trying in vain to copy the first section of unreadable sectors.
I believe there are several tutorials for the process that you can find with google. You would need a reliable hard drive with the same or greater capacity as the failing hard drive to clone to. I suggest having no other hard drives connected to the computer when cloning to eliminate confusion and mistakes. You would want to make absolutely sure that when you run the command you do not have the two devices (hard drives) reversed, otherwise you would copy the new hard drive to the old, effectively erasing it. Once the initial copying of easily readable sectors is finished, it may take a ridiculous amount of time to read whatever usable data may exist in the more problematic sections. Many people likely just stop the process there since you don't get much return from that point.
It is possible, but less likely, that the hard drive itself is fine, and you just have some data corruption in the file system. If that were true, you'd probably still want to copy off the data so you could run Microsoft's chkdsk to try and fix it without having to worry about chkdsk screwing it up even more. A program called smartctl comes with most linux live disks, and it can tell you if the hard drive itself has problems (reallocated sector count is the biggest clue).
