If you have already read about lighter than air vehicles (LTAV) in the 1930s, you know that they are very vulnerable to weather. This is why, to my understanding, they were abandoned.
However, predictive weather radar has improved significantly in resolution and accuracy. I can imagine the floating civilization always moving to stay in the clearest part of the weather track.
Interactive Future Radar
Another concern is air envelope weight. I'm going to imagine you using handwavium for your envelope : maybe some next-generation waterproof weaveable aerogel. Even neglecting infrastructure, depending on the height of your floating civilization, you will need a very large displaced air volume.
- 1,000 feet: 1,700 cubic feet per hundred pounds
- 5,000 feet: 2,000 cubic feet per hundred pounds
- 10,000 feet: 2,500 cubic feet per hundred pounds
The altitudes at which an LTAV city would work physically are in the middle of the aviation range of altitudes. Internationally, certain paths have published altitudes and there are levels of separation between traffic. It may be possible for your civilization to negotiate to have a Notice to Airmen (NOTAM) issued directing flights away from the city.
I imagine most nations would want to refuse a request for a city to float over their land. However, if the city is organized by an institution that does business and has leverage almost everywhere (think of an Exxon or other multi-national energy company), then I can imagine them being able to negotiate fly-over requests with most countries.
If the city was an international station, like the International Space Station or international antarctic stations, then it seems plausible that universities and governments would provide most of the operating income for the station.
As a suggestion to what kind of research a floating city might accomplish: they could focus on climate research or climate change research, or maybe are evaluating the effectiveness of political/technological improvements that are supposed to reverse climate change.
The city doesn't have to be the best solution to the problem. We live in a political world where the best solution is often not the one we go with.
Replacement helium will likely be your most frequent material cost. At $450 USD per 1,300 cubic feet according to Quora your cost would be:
1,000 ft: $575 per hundred pounds of weight being carried by the city
5,000 ft: $680 per hundred pounds
10,000 ft: $865 per hundred pounds (all values rounded)
According to this YouTube answer, and assuming your LTAV helium envelope was made of something similar to Mylar in impermeability, the civilization would need to replace the whole volume of helium approximately every second week. That would make annual helium cost $15,000/$18,000/$23,000 USD for the three altitudes mentioned before. Multiplied by the weight of the rest of the settlement in hundreds of pounds.
Assuming the flying city had the $125 million per year maintenance budget of the International Space Station, how big could the aircraft based civilization be? Between 418 tons at low altitudes to 280 tons at the higher altitude.
How Much Will Infrastructure Weigh
Again, excluding the helium envelope, it would be nice to know what is required to sustain residents at this high altitude civilization. It will be cold and air will be thin. A good example of structures built for this purpose would be the McMurdo Station modules in the Antarctic. These had to be able to do the job, but also be small enough to fit into the 30,000 pound footprint of a C-130 delivery vehicle.
Given an International Space Station maintenance budget of $125 million per year to spend on helium, you could have between 18 and 27 McMurdo Station style modules for your floating civilization.
You would probably want to cut out an allowance for aircraft to arrive and depart the floating city. Even at 10,000 feet altitude, the aircraft based civilization is within reach of propeller aircraft. A few aircraft weights for reference: a six-passenger Learjet 24 is about 6.5 tons. An Airbus A300 (323 passengers) weighs about 70 tons.
Also, you would want weight for people and supplies.
From a hard-science perspective, you might really want to think through it. Especially the material holding all the helium keeping the city afloat. It would be very easy to drive up weight and cost there.
It is realistic that someone motivated with a will to make it happen and good international connections could make such a place possible. Yes.