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Do not use the drive at all until you start the rescue operation!
I suggest that you try ddrescue.
Install and run it from a separate drive, the boot drive (for example a persistent USB pendrive made from an Ubuntu desktop iso file).
Read carefully the instructions at the info page
sudo apt-get install ddrescue
Example 1: is probably the best starting point, but you must change the device names to fit correctly your case. Use what you find from
Check and double check! If you get it wrong you might overwrite and destroy your data instead of saving them.
sudo parted -l
Consider the present drive as the source. And get another drive of at least the same size an use it as the target drive. Then use the method to clone the source drive to the target drive.
Example 1: Rescue a whole disc with two ext2 partitions in /dev/hda to
Note: you do not need to partition /dev/hdb beforehand, but if the
partition table on /dev/hda is damaged, you'll need to recreate it
somehow on /dev/hdb.
ddrescue -f -n /dev/hda /dev/hdb logfile
ddrescue -d -f -r3 /dev/hda /dev/hdb logfile
e2fsck -v -f /dev/hdb1
e2fsck -v -f /dev/hdb2