First step, in case you didn't do that yet: Create a disk image of the partition - you don't want to try data recovery on the actual data. Easiest is just using dd to dump the disk to another drive.

Next try running testdisk on the image to see if it can find the backup superblocks - if it does you can feed that to fsck to restore the filesystem.

If you know the blocksize of the filesystem you can also run mke2fs with the -S parameter - this will just write the superblocks. Again, only do that on a disk image, not the actual drive.

Try testdisk. It can copy files from damaged filesystem without touching it. But only if you are lucky enough and the filesystem is not so heavily damaged that testdisk will be unable to find it.

As others have said, first make a full copy of the disk so you're not working on a failing drive.

Then, I highly recommend DDRescue

The first thing you should do is create a raw backup of the data. The more you use this drive the worse the failure gets.

The quick and dirty way I've used is...

Use the nbd system (network block devices) and qemu to create a qcow2 image with your defective device as the base device. Serve this qcow2 image with qemu-nbd and attach it as a NBD device locally. Then run fsck or testdisk on the NBD device. This will let you repair the filesystem Linux sees without writing to the disk. Testdisk can scan for any filesystems left on the device if the partitions no longer match filesystems.

Also, if all else fails use photorec to slice the file types you need.

Also, ddrescue can try to read any actually failing sectors and work out what they contain, but puts a lot of stress on the device.

Beware, any method that puts more wear on the disk should not be used unless you're willing to accept the risk that the drive could get worse.

First create an image (cat /dev/your-disk > /path/your/image) and use recovery tools on there.