For me, at least, this is a fascinating article.
Tom Persky is the self-proclaimed “last man standing in the floppy disk business.” He is the time-honored founder of floppydisk.com, a US-based company dedicated to the selling and recycling of floppy disks. Other services include disk transfers, a recycling program, and selling used and/or broken floppy disks to artists around the world. All of this makes floppydisk.com a key player in the small yet profitable contemporary floppy scene.
The American nuclear silos use computers that were made in the 8-inch floppy disk era, and they’ve had to obtain replacement on eBay. The computers are designed to launch nukes, and nothing else, and AFAIK are not hackable because they aren’t connected to anything but a self-contained electricity supply.
What in the world could you possibly use them for these days, outside of weird un-updated systems like the Minuteman launch systems? Almost no computer files these days are small enough to fit on a floppy anymore- the photos from my phone are roughly 3 mb each which is twice the size of the 1.44 mb floppy. And of the 1500 mp3 files I have, only 36 are small enough.
Honestly a flash drive is faster, larger capacity and easier to use. There’s no practical reason to fool with them anymore; my last 3 PCs (I replace/renovate them about every 6 years) haven’t had floppy drives, and I haven’t missed them.
Documents? Even some PDF’s are under 1 MB. And plain word docs are usually just a few KB.
And I just checking some of the pictures that I’ve downloaded from online news articles, etc. – many of them are less than 1 MB.
My wife (soon to be Dr. My Wife in 8ish weeks) had to salvage a 5.25 floppy drive and disks in order to use some of her equipment at school. Until last year, 3 separate systems used 5.25, but after some upgrades, it’s now only 1.
But quite a few still use 3.5 disks. To the point where I had to buy her a 3.5 external drive with a USB interface to make it easier to work on some of her research at home.
But she’s more upset about the device that was designed to output data to its custom printer. So it literally will not work without the printer on and attached. Or it’s custom keyboard, or any of it’s other bespoke equipment that hasn’t been made in 3 decades.
The article nowhere says that people are using them to store new documents (I mean you could… maybe some people still like to write their novels on a Mac Classic)
But if you have a machine still in service that can’t yet be replaced that uses, say, 8-inch floppies, you have to get them somewhere… (or be prepared to do some real hacking)
Reminds me of the time I had to read an old 9-track(?) tape. (Just read it; I wasn’t going to use a new one.) It was fun trying to find someone with a drive…
Hijack: Congratz
We still use CDs at work to transfer files between computers. (Flash drives are banned, and hard drives take forever to get approved for use.)
Floppies are still in use in Japan. The government recently “declared war” on their use.
There’s also a pretty decent retro computing scene, where having access to new disks can be nice. (The alternative involves using hardware disk emulator that use an SD card or similar but pretend to be a disk.)
Why specify the “outside of”? There are a lot of weird un-updated systems out there. Surely enough to make for a business niche for a single small company.
Thank you @Saint_Cad, I’ll pass it along!
But back to the OP, when my wife was at Boulder (CSU sister school), which is MUCH better funded, they too had plenty of obsolete data storage systems in use. A lot of that is Universities are often getting ‘gifted’ with partially functional 2-3 generation old tech which is still a godsend considering the cost, but keeping it in parts/equipment is a logistic nightmare. And floppies, which are inherently prone to damage due to the mechanical reading, are one of the worst.
Many a day do the grad students and faculty spend on Ebay and electronic stores searching for parts to repurpose. Or going through storage rooms to scavenge old power supplies, drives, and the like. For that matter, we personally lent our old but functional HP1020 laser printer to the office - because a lot of the lab equipment wouldn’t talk to the modern era of wireless only printers! 
Until about five years, the machines they used to examine my pacemaker every six months used floppies and gave them a whole pile of them after I found out that they had trouble finding them. They have now upgraded and use USB sticks.
I remember in high school, we all carried around small blue floppies for our English documents.
If you do not literally need floppies, you can always hack up a device that plugs into the same interface but uses usb sticks or SD cards or whatever for the actual storage. You do not even need to be such a hacker, since chances are someone already did the hard work for you.
Another approach is to emulate the whole system, software and hardware. If it is old enough then the emulated system should run at least as fast. But for laboratory instruments I do not see why you would not eventually upgrade.