Here's an app that promises 4K video from your iPhone 5 or 5S by, presumably, doing some crazy photo burst mode trickery and then stitching the results together at the end. Interesting.
If my math is correct and a single image from an iPhone 5 is approximately 2.3MB, that means one second of 24fps footage will take up about 55MB, maybe more if there's a lot of color/motion information; one minute of this footage will take up 3.3GB of space. Subtract about a gigabyte of space that iOS itself takes up and here are your shooting times per device:
Keeping in mind these are *best case* estimates, you'll very likely get less time to shoot. This program also offers a rudimentary editing interface as well as a way to upload directly to Vimeo, YouTube, Facebook, etc.
It costs $999.
Let that sink in for a moment.
$999. For an app, specifically for iOS 7 at the moment, that lets you, for some reason, shoot 4K on your phone.
Let Me Offer an Alternative
If you happen to have a cool grand lying around the house that you just **have** to spend on something video related, might I suggest you wait just a touch longer and spring for a newly-announced Panasonic FZ1000. It shoots 4K in a slightly more manageable format with better compression, comes equipped with a Leica lens that allows you to zoom from 25-400mm, and has better ergonomics due to form factor.
Additionally, you could edit this footage in Premiere or Final Cut Pro X, both of which are far more robust than anything your phone currently offers. If you have a reason you need to be shooting 4K, you should already probably be using one of these platforms.
I'm not saying that buying this app for $999 is a bad idea, I'm saying it's a terrible idea. The device itself is not built to shoot this way, 4K offers no advantage to a person not heavily involved in editing or in possession of a 4K television, and there's no guarantee Apple won't close this obvious loophole in a future software update.
My advice would be to stick to shooting HD for now and up your 4K game when it's built into iPhone 7... and have at least 128GB of storage on your phone.