Error 6003F

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Start your MacBook holding Command, Option (Alt) and R keys while your computer is starting, and check that you can start in Internet Recovery. Then, open Disk Utility, choose Macintosh HD in the sidebar and verify the disk. If it's damaged, repair it. If it was repaired, close Disk Utility and reinstall OS X. If it couldn't be repaired, take your computer to an Apple Store or reseller.

If you can't start into Internet Recovery, take your MacBook Air to an Apple Store or reseller

I had this same problem, and despite the message, 6003F is not documented anywhere. Unfortunate.

I believe this simply means the Apple servers aren't visible on your default network—in this mode, your Mac can't access your saved network name and password. If you hold the Option key on bootup instead, you'll get the Startup Manager. That screen will let you select a wifi network and enter a password, if needed. I did this and was able to download the recovery image.

Good luck. I'm answering all the recent 6003F questions I can find to help the next person 🙂