My MacBook Pro has a flashing question mark in a folder against a grey background My MacBook Pro shows a grey screen with a flashing question mark in a folder.
I tried connecting it to an older MacBook using firewire cable. I restarted the Macbook Pro in Target Mode. Now I see Thunderbolt and Firewire icons flashing and moving in a grey background. There is a green power icon at the bottom as well.
I am not an expert even though I have been using Macs for ages.

How can I fix this issue so I can recover/backup things in the MacBook Pro hard drive?
Either the Mac OS X installation is corrupt and won't startup the computer or your hard drive is faulty, physically.

You could try to connect another Mac to your mac via Firewire or Thunderbolt in Target Disk mode to actually see something. 
That's what the Target disk mode is made for.

You could also try booting from the original DVD/USB, then run Disk Utility. If you can see the hard drive in the list, try repairing the disk.