An error has occurred while erasing your restore destination disk

Question

MacBook Pro, mid-2010 15-inch; OS X Mavericks (10.9.4); originally shipped with Snow Leopard.
My MBP has crashed & won't start beyond the grey screen and spinning wheel. System restore from Time Machine backup won't work. I tried all these without success
Safe mode (hold Shift during power-up)
Verbose safe mode (hold Shift + Command + V during power-up) which reports
           ** /dev/rdisk0s2 (NO WRITE)
               ** Root file system
                    Executing fsck_hfs (version hfs-226.1.1).
               disk0s2: I/O error
          ...followed by some other stuff I haven't transcribed before it grinds to a halt
Reset NVRAM (hold Option + Command + P + R during power-up)
I can boot up ok into the Recovery partition (hold Command + R during power-up) to run Disk Utility and Time Machine.
I can’t start from the OS X Install disk (hold C during power-up) cos my DVD drive spits out all CDs and DVDs after spinning them up. This despite cleaning with compressed air.  Tried to run AHT, the Apple Hardware Test (hold D during power-up): nada. Also running AHT from the internet - (hold Option + D during power-up), still nothing.
I’ve verified my hard disk (all appears ok) and run Repair Disk Permissions.  This reports some problems, for example:
        Permissions differ on “System/Library/CoreServices/Feedback Assistant.app”; should be drwxr-xr-x ; they are lrwxr-xr-x .
          Unable to set owner and group ; error 22: Invalid argument
          Unable to set permissions ; error 22: Invalid argument
Same problem (but with various different permissions) for
          “usr/lib/libruby.2.0.dylib”
          “usr/lib/libruby.dylib”
          “Applications/Safari.app/Contents/Resources/Safari.help/Contents/Resources/inde x.html”
Also have lots where as well as unable to set owner, group and permissions the
          Group differs; should be 80; group is 0.
Affected items are
          “Library/Printers”
          “Library/Printers/Icons”
          “Library/Printers/InstalledPrinters.plist”
and a few dozen more for various items in subfolders of “Applications/iBooks.app/Contents”
I don’t know if they’re linked with my problem or an irrelevant side-issue.  I don’t understand the ramifications of these wrong permissions that Disk Utility can’t repair.  And I don’t have the headspace or time to learn now either.
I repeatedly tried system restore from Time Machine backup, but get the error:
          An error has occurred while erasing your restore destination disk.  Restart your computer and then try restoring again.
I tried erasing the hard disk partition with Disk Utility but got the error
          Volume erase failed with error couldn't unmount disk.
Now totally out of ideas.  Have I overlooked anything?  Is my hard drive a goner?  BTW my battery is also saying "Replace soon", but I don't know if that's relevant.  All help gratefully appreciated!

Answer 1

There are several ways to back up a Mac that is not fully functional. You need an external hard drive or other storage device to hold the data.
1. Start up from the Recovery partition, from Internet Recovery, or from a local Time Machine backup volume (option key at startup.) Launch Disk Utility and follow the instructions in this support article, under “Instructions for backing up to an external hard disk via Disk Utility.” The article refers to starting up from a DVD, but the procedure in Recovery mode is the same. You don't need a DVD if you're running OS X 10.7 or later.
2. If Method 1 fails because of disk errors, then you may be able to salvage some of your files by copying them in the Finder. If you already have an external drive with OS X installed, start up from it. Otherwise, if you have Internet access, follow the instructions on this page to prepare the external drive and install OS X on it. You'll use the Recovery installer, rather than downloading it from the App Store.
3. If you have access to a working Mac, and both it and the non-working Mac have FireWire or Thunderbolt ports, start the non-working Mac in target disk mode. Use the working Mac to copy the data to another drive. This technique won't work with USB, Ethernet, Wi-Fi, or Bluetooth.
4. If the internal drive of the non-working Mac is user-replaceable, remove it and mount it in an external enclosure or drive dock. Use another Mac to copy the data.