I think my HD is failing. Can't backup files from recovery mode. How can I preserve my data?
I had this problem too, the best thing you can do is replace the harddrive since it will eventually break beyond repair.
Macs are quite sensitive to faulty hdd's so there is a good chance you can still use the drive as external drive and get your data from it.

so:
Replace the drive yourself if you can (ifixit) or have it replaced
Get a hard drive enclosure for your old drive
plug it in your imac and drag back the files you still need

Once you've removed the failing/failed drive, you can then use an inexpensive USB to SATA adapter and see if any of the off-the-shelf file recovery software can read some of your data from the drive. Data Rescue might be a good tool to start with since the free demo will tell you what it might be able to recover before you have to spend $$ on the software.

Lastly, sending the failed drive into a professional will cost $$$ but if you really need the files or pictures back, that's often the best chance of success.