Gray screen on start up and not able to repair hard disk and not able to take back up.

Question

My macbook pro bought in early 2011 with snowleopard software and than upgraded to lion is now running on system 10.7.5 .
My problem is i get a gray screen on startup with apple logo and spinning ball.
I tried all the solutions mentioned in the apple support for "gray screen appears during start up". This Disk utility says that the Hard disk cant be repaired.
Before i format my disk and erase all data and install Mac os x again, i want to take back up on a external hard disk. I bought a new 2tb external hard disk so that i can take a back up.
As i try to take the backup of the image, it gives me an error "cannot allocate memory"
i tried to change the configuration from compressed to read/write and tried different encrption methods but unable to take backup.
i sometimes get error "input/output".
how do i take a backup. somebody please help

Answer 1

See OS X- Gray progress bar appears under Apple logo during startup. If you need to reinstall OS X, then do this:
Reinstall Lion, Mountain Lion, or Mavericks without erasing drive
Boot to the Recovery HD:
Restart the computer and after the chime press and hold down the COMMAND and R keys until the menu screen appears. Alternatively, restart the computer and after the chime press and hold down the OPTION key until the boot manager screen appears. Select the Recovery HD and click on the downward pointing arrow button.
Repair
When the recovery menu appears select Disk Utility. After DU loads select your hard drive entry (mfgr.'s ID and drive size) from the the left side list.  In the DU status area you will see an entry for the S.M.A.R.T. status of the hard drive.  If it does not say "Verified" then the hard drive is failing or failed. (SMART status is not reported on external Firewire or USB drives.) If the drive is "Verified" then select your OS X volume from the list on the left (sub-entry below the drive entry,) click on the First Aid tab, then click on the Repair Disk button. If DU reports any errors that have been fixed, then re-run Repair Disk until no errors are reported. If no errors are reported then click on the Repair Permissions button. When the process is completed, then quit DU and return to the main menu.
Reinstall Lion, Mountain Lion, or Mavericks
OS X Mavericks- Reinstall OS X
OS X Mountain Lion- Reinstall OS X
OS X Lion- Reinstall Mac OS X
     Note: You will need an active Internet connection. I suggest using Ethernet
                 if possible because it is three times faster than wireless.