i went into recovery mode and tried to repair my hard drive and it says it can't be repaired and I have to restore my computer. is there anyway i can backup my files from recovery mode onto a new disk?
iMac
Not that I know of; you would need plug in an external drive and backup your files there - either by drag 'n drop, Time Machine, or a bootable clone. Recovery mode isn't designed to initiate a backup. If you cannot boot into your hard drive, boot from an external source and try to access your hard drive from there.
There are several ways to back up a Mac that is not fully functional. You need an external hard drive to hold the backup 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, and no other Mac is available, 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.
I need to amend my first answer: I just learned that you should be able to use Disk utility while in recovery mode to make a backup; however, it will be an all or nothing thing and you need to have an external hard drive attached of course. Just follow Linc's directions.
How can i backup my files in recovery mode?
