Flying aircraft carriers are staples science fiction and steampunk. It's an idea that sounds like it would be completely crazy in reality, but it's not crazy enough to keep the Air Force commissioning a report on the idea from Boeing back in 1973.
This wasn't a new idea, even in the 1970s. The Air Force had tried to make it work several different ways, starting with F9C Sparrowhawks that could be launched and retrieved from airships like the Macon and Akron, which was sort of a success. Better luck was had with launching small fighter aircraft from bombers (as with the XF-85 Gremlin), but the idea was still deemed too risky, since docking with the bomber in-flight was almost impossible, even during the day in good weather. Eventually, the Air Force just gave up on the idea entirely and made due with mid-air refueling, which restricts aircraft to single missions with single payloads but removes range constrains due to fuel capacity.
By the 1970s, though, the Air Force was in need of "a versatile system with global range and supersonic performance" that could provide for a 24 hour strike capability anywhere on Earth. One option would have been to build air bases in Europe, Africa, and Asia, but the Air Force thought that it might make more sense to build mobile air bases instead, so it asked Boeing to take a look and see if turning 747s into aerial aircraft carriers was a possibility.
The difference between an aerial aircraft carrier (AAC) and an aerial fuel tanker is that an AAC can rearm aircraft as well as refuel them, crews can switch out with each other, and it's even possible to perform repairs and maintenance on fighters. So obviously, an AAC would be pretty useful, but is it realistic? According to Boeing, definitely yes.
Boeing took their 747-400 and hollowed out the inside, leaving two decks worth of open space. It didn't make any significant structural modifications to the 747 airframe, meaning that fighter aircraft would have to fit within just over 17 feet of width. This, of course, necessitated designing an entirely new "microfighter," and Boeing drew up five concepts:
These microfighters would have been tiny. A loaded F-16 weighs about 40,000 pounds, while a loaded microfighter was just over a quarter of that. They weren't designed with landing gear, and since they didn't have to climb up to altitude, they didn't need to carry nearly as much fuel. As for armament, the plan was to give them a pair of 20mm cannons, along with pylons for smart bombs and air to air missiles.
The interior of the 747 would have been split into an upper hangar deck and a lower flight deck. Full complement was ten microfighters (!) plus a crew of 42. The flight deck had launch and recovery bays both fore and aft of the 747's wings. Two microfighters could be prepped and launched at once, and all ten could be launched in intervals of just 80 seconds a piece. Ten minutes was sufficient time to rearm a microfighter from scratch, and enough supplies were carried onboard the AAC to allow each fighter to fly three separate missions. Meanwhile, a crew rest area and lounge let pilots rest and recuperate.
Now, the really tricky part would of course have been the microfighter's docking maneuver. By this time, fighter pilots were getting halfway decent at midair refueling, so Boeing figured they'd just work the docking mechanism into the refueling system. A microfighter would hook up to a refueling boom underneath the 747 and fill its tanks in about 30 seconds. Then, the boom would retract in towards the hangar bay, with the microfighter still attached, and a trapeze device would swing down and grab the microfighter and hoist it up into the hangar bay with the pilot completely hands-off. A hatch would close, the flight deck would pressurize, and then the microfighter could be rearmed and relaunched or swung up into the hangar deck.
The AAC was not intended to work alone. In fact, Boeing envisioned a "Multi-Purpose Strike System" utilizing a fleet of ten 747-AACs all overseen by an AWACS. This would enable 100 fighter aircraft to be deployed to anywhere Europe on eight hours notice, which sure sounds good on paper.
Boeing made sure to point out that "no unique technology evelopment has been identified for Airborne Aircraft Carrier. Demonstration of capability is possible within the current state-of-the-art." In other words, Boeing thought that it was certainly possible to build an AAC with the technology that existed at the time (or at least, that would exist by 1980), and it recommended detailed design studies along with preliminary modifications to a 747 that would have allowed for testing of a docking system.
So what's the problem? Why don't we have 747 Airborne Aircraft Carriers? Well, "technical feasibility" and "the potential of great national benefit" (Boeing's words) is one thing, but practicality is something else entirely. The AAC would have been a very big, very expensive risk that didn't offer enough unique benefits (relative to naval aircraft carriers and land bases) to justify development, and the microfighters were too specialized to be able to fulfill the AAC's promise of near-instant versatility.
That said, it's entirely possible that there might be a new niche where an airborne aircraft carrier could thrive, and that's with UAVs. UAVs already know how to land on other UAVs, and if we can make the whole shebang autonomous (with, say, a solar-powered high-endurance UAV as a mothership), we'd have something almost (but not quite) as cool as the 747-AAC.