The following answer by Ivan Bartsov is copied from Unix&Linux Stack Exchange

Ext filesystems store backups of the superblock -- for an occasion just like this one.

First, determine the locations of the backups (make sure you have the -n option! otherwise this will wipe the file system with a new one):

mke2fs -n /dev/sdxx

This is a test run (i.e. no write) of FS creation routine. It will let you know the offsets of where it would put superblock backups if it were creating a filesystem. If you know your FS block size to be something other than 4096, you must also specify the parameter -b {blocksize} to get the right numbers.

For 4096-sized blocks, first backup superblock will be at 32768. If the following operations fail with the same error message about bad superblock/short read, try the next superblock backup from the list mke2fs gave you.

Next, you can either mount the filesystem using the backup superblock like this

mount -o sb=32768 /dev/sdxx /mnt/sdxx

then explore the FS from your file manager, copy undamaged files etc.

Or, to actually fix the FS, you can run fsck with backup superblock like this

e2fsck -fy -b 32768 /dev/sdxx

Here -f makes it scan the disk even if it's not dirty and -y answers yes to all enquiries about fixing stuff. The -b option, aside from specifying the backup superblock, will make it update the original superblock with info from backup.

After this, you should have your filesystem back.

If e2fsck fails to write the main superblock If the superblock got corrupted because it's on a bad sector e2fsck will finish the run, attempt to update the superblock, and give you the following error message:

Error writing block 1 (Attempt to write block from filesystem resulted in short write)

Obviously, the main superblock isn't updated and the whole e2fsck run is in vain.

You need to hint the disk to remap that sector -- by writing zeroes to it. Thanks to @Keith for pointing this out: the next command can make a lot of mess if mistyped, so triple-check it before running. Here's the magic:

dd if=/dev/zero of=/dev/sdxx bs=4096 count=1 seek=0

This will write 1 4096-sized block of zeroes to sdxx at offset 0. Don't forget to account for different block size if that's the case for you.

After that you'll be able to write to superblock (which will be on a different physical sector transparently to you). Now, you run the e2fsck command above again and it should succeed writing the superblock allowing you to mount the FS normally.

Goes without saying Now you should backup the critical data to another physical drive and, if you still plan on using the filesystem, run

e2fsck -fccy /dev/sdxx

PS Kudos to @Nemo on this find: in case all of the backups of the superblock of your FS are corrupted, mke2fs/mkfs has the -S option that will re-create the superblock and group descriptors as if creating a new filesystem, without touching anything else. But you absolutely have to be sure your blocksize is right and the man page says you should run e2fsck after it and there are no guarantees about data being left for rescuing. Read the man page and throw a plus on this answer.