Did you set up your contacts to sync with Google? If so, they've been backed up automagically. If you're not sure, log in to your Gmail account (the same account you used to set up your phone) on a computer browser. In the top left, click where it says "GMail", then choose "Contacts." See if all of your phone's contacts are stored there. If so, they're be restored to your phone when you get it back from Google.
If your goal is to delete the contacts, to make sure no one at Samsung can see them when they repair your phone, that's probably doable by doing a factory reset, if you have the stock ROM and recovery. This video:
How to enter Android recovery & Factory reset the Galaxy S3 - YouTube will show you how to do it. I know you can't see anything on the screen, but this uses the physical buttons to enter recovery mode and reset, and those should still work. You'll just have to count button presses instead of watching on the screen.
If you want to copy off the contacts, and they're NOT synced with Google, I'm not sure I have an answer. If you take the phone to a carrier store, they might be able to do it with the tools they use to copy use data to a new phone.