Trouble backing up HDD in Mountain Lion Recovery Mode into external hardrive

Question

My Macbook Pro that's only more than 3 months old is stuck in the grey screen with the apple logo, so I'm stuck in OS X Recovery Mode. While I've tried ressurecting it using safe boot or the fsck command, it's simply not working. I've decided to just reinstall the OS X, but I think there's still something wrong with my HD that's preventing the OS to be reinstalled.
I'm planning on doing an Erase & Reinstall AFTER I back up my HD to one of my two external hardrives. But while on disk utility & using the Restore tab to copy the contents of my HD to my external hardive, it says that it "Could not validate source - error 254". When I tried erasing the contents of my external first, it says "Disk erase failed. POSIX reports: the operation could not be completed. input/output error."
What should I do in order to be able to backup my HD to one of my external hardrives?
WIll erasing/reformatting my HD let me reinstall OS X via Internet Recovery mode?
Please help! I'm really hoping I'd be able to save my files without resorting to spending cash for file recovery programs. And I need to be able to use my laptop again soon.

Answer 1

Since youre not at all familiar with doing that, I wouldnt recommend you doing that. 
Make a service call to Apple for diagnostic / repair.     Are you under warranty?
Well, your laptop is ONE thing, the HD is another.   There are only 4 HD makers on earth, and all the pros work from the premise that their HD is going to crash and fail at anytime, all the time.  Even the very best made ones.
Consider these 2 points for the future:
Never consider any computer a data storage device at any time under any circumstance, rather a data creation, sending, and manipulation device. Anyone who thinks data is safe on any computer, even copied upon multiple partitions is making a mistake that will, without fail, strike.
Most importantly know that 2 copies of your data is 1, and 1 is none.
“My computer is OK, therefore so too is my data"
Literally thousands of times, regardless of make or model of computer I have heard “my data I have spent months on is all gone”; or likewise “my computer with priceless data was stolen/in a fire and the backup (singular) was with it and is gone too”. When we fail to plan, we are planning for certain failure in neglecting data redundancy. At the very minimum of $10,000 a month, anyone who has lost ‘months’ worth of data has trusted a very valuable payload, sometimes $100,000+ of work at the end of a very weak chain.
How so very weak? All HD/SSD, even the very highest of quality, are at best considered ephemeral by professionals and a single backup is merely a rope in the case of a ‘fall’, as in the case of drive failure. In addition to a rope your data needs a net and additionally a cushion underneath it all for safety to be considered in place.
None of us would consider working hard for months and then taking the money from that work and placing it on the curb in a paper bag; yet most people never second guess leaving that level of work on our computer drive or a single backup. Separation in place and unassailable redundancies is mandatory. It takes a matter of mere minutes to safeguard priceless data, therefore proper consideration and action needs to be taken.