I guess, being National Geographic this probably was a massively expensive drone and camera, but there are cheaper, lighter options available to the rest of us, and they've been around for a good 2 or 3 years. The smallest of them could be drone-mounted...
Here's someone hanging an older Ricoh Theta from an Inspire 1.
Terrible quality footage, but it proves it's possible, and the newer Theta V does much higher resolution so video is correspondingly less awful, though still up to the sort of reoslution we;d be looking for. 4K isn't so good when it's spread out over 360 degrees...
Here's the rather better but less helpfully shaped 360fly, which is based on a sphere, but does have mounting options and remains fairly light.
DJI has it built in on some of their better Drones as a add in Spark , Mavic or better. It's called Panorama.
I Run the Photosphere Community on Google +. And I have had people posting 360 drone videos for awhile now.
It's just a matter of learning how to edit the video or picture to patch the sky' since most drone cameras can't shoot up. Doing a search on your make and model will tell you if the plugin is available.
You don't need the Fly or a Theta hanging to create these.
Also doing a search on YouTube Keyword 360 which has been around almost 2 years now will give you thousands of results.
If you have questions I worked with Google many years ago on 360
To add to this you can post these type of video to Facebook and other types of media.