A failing drive needs to be imaged. Once it is imaged, you can mount it as a regular drive, or if the filesystem is damaged, you can pick the files out using file-carving.

The Rescue-Remix live system is command-line only. If you are uncomfortable with this, you can run the live cd, connect to the internet and install the rescue-remix tools.

You need to use gddrescue to image the drive. Basically, mount a drive somewhere else which will be big enough to hold the image (the entire contents of the drive, even the empty space).

Now, say you mounted that drive in /media/Disk/ and the failing drive is sdb, you would run the following:

sudo ddrescue /dev/sdb /media/Disk/image /media/Disk/log

If you only want to recover the first partition (or if there is only one partition on the failing drive), then do it this way:

sudo ddrescue /dev/sdb1 /media/Disk/image /media/Disk/log

If that completes and there are still errors, make it go over the bad spots a few more times:

sudo ddrescue -r 5 /dev/sdb1 /media/Disk/image /media/Disk/log

Once you have done this and you cannot get any more data back, try mounting the filesystem

mkdir mnt
sudo mount /media/Disk/image mnt

If that works, then browse the mnt folder you just created (example: /home/Ubuntu/mnt - NOt /mnt) and copy the files you need to rescue over to another drive.

If you cannot mount the image, it's filesystem is corrupt, so you need to use file-carving.

sudo photorec /media/disk/image.

For more detailed help, see the Data Recovery wiki page on the help.ubuntu.com wiki (the link found on the Documentation page here)