i think your hard disk is fried
i did not even look up what those errors meant, but from the look of the screen , i think it means your hard drive is completely shot and it needs to be replaced
you did a real good job of keeping it backed up to time machine
replace the hard disk, and reinstall mac os x.
how to make a bootable usb key .
have someone at the apple store or another person with a mac help you
http://www.macworld.com/article/236...otable-os-x-10-10-yosemite-install-drive.html
check warranty status
maybe its under warranty or you have applecare.
https://selfsolve.apple.com/agreementWarrantyDynamic.do
once mac os x is reinstalled, you can restore it from time machine
http://support.apple.com/kb/PH14111
if the hard disk was purchased from another source except apple, like if you installed this disk yourself, please write back. it might be covered under a separate warranty from the original drive manufacturer
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once you find a bootable usb key and use that to boot your computer (either by holding down the option key or holding down the c key ) it might be possible that disk utility will fix that,, but i wouldn't even bother. it looks totally shot to me , i don't think disk utility would fix it and you should throw out the had disk.
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after reading the screen your hard disk is shot and it is reporting bad sectors.
replace hard drive
there is no other real diagnosis or utilities you should run. its dead and put a fork in it, and buy a new hard drive.
however
if this hard drive was a drive you put in, you need to get the driive utilities from the hard disk manufacturer and give them the error codes to have the hard drive replaced under warranty.
these programs are free to download. they have names like
SEATOOLS
Hitachi DFT
Western digital lifeguard diagnostics.
don't spend any money on "s.m.a.r.t. checkers." they are useless. your paying $$ for an application that is freely given away by the manufacturer of the hard disk
you only need these utilities ( and probably a windows machine ) if the drive was purchased separately and installed by you or someone else, and that hard disk came in a retail box
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every single hard drive manufactured since IDE came put ( or what you guys call PATA which was before SATA ) , this would be cover about 25 years i would say, uses automatic error correction and detection. how this works is if the hard drive finds a bad sector, it moves the data to a spare sector, and does not inform the host operating system. it just fixes itself and never tells you anything. all defects on the hard drive when it was shipped are hidden to you. read write errors are automatically corrected without you knowing about it
when your operating system reports a bad sector, it is because the hard drive has ran out of bad sectors, and it is no longer reliable, and it is shot and it needs to be replaced.
further more, hard drives have "s.m.a.r.t". once the derive says this smart has failed, the drive is considered shot.
if you boot a usb key, you might be able to check the smart sensor by clickng on about this mac , system report,. but i from looking at the error screen, i think that would be a waste of time , and i would just replace the hard drive,.
if your computer is under warranty or appecare, apple will be glad to replace your hard disk
if your no longer under warranty you need to buy a new hard drive
if you bought that hard disk in a retail box and then installed it in your computer, then you need to get those previously mentioned utilizes, write down the serial # and the failure code and get the manufacturer to replace the drive under their warranty