You're joking, right?
Almost every computer that handles classified information for the DoD is connected to a network. Not the Internet of course, but SIPRNET or one of the 30 or so other classified networks, depending on classification level and other considerations. I don't recall ever needing "a key, a passcard, and supervision" to access any of them, just a user name and password, like every other computer.
Damn near nothing is paper only anymore, and any time I needed a copy of a document I clicked the "print" button in Word or Acrobat, walked over to the printer and grabbed it. And yep, I can email them too! Only to accounts on the same network of course, and I am ultimately responsible for determining whether or not the recepient has the appropriate clearance and need to know, but it's that simple.
Lots of things are sensitve but unclassified (also known as SBU).
I hate calling people out like this, but you're spouting lots of FUD. You're either intentionally lying about how things work, or you've never had any contact with anything classified and are rattling off garbage that you either made up or pulled from some crappy novel.