I know the number 1000 is just an arbitrary number that happens to have some zeros in it, but is there any non-economic obstacle to making a 1000 passenger aircraft?
Also, assume normal-ish space-per-passenger constraints, so 1087 people on a 747 isn't really a 1000-passenger aircraft.
Yes, there are some barriers:
First the scaling laws: When you increase the size of an aircraft, its area will grow with the square of the increase while its volume will grow with the cube of the increase. This demands proportionally bigger cross sections of all load-carrying members. For the same reason ants cannot be scaled up and elephants have much bigger legs than antelopes. The A380 is already pushing the envelope, and any bigger aircraft will be even more problematic to design. But not impossible.
The emergency slides on the A380 were one of the most challenging parts to design. While normal slides can be mercifully short, lowering a passenger by 8 meters from the upper floor such that he or she will reach the ground at a survivable speed is hard. The slide must be much longer and still is not allowed to buckle. This required the development of new materials.
Now that this has been done, a stretched fuselage with another two or four type I exit doors will make the 1000 pax version possible. But when a future design plans to have a wider fuselage with more seats in a row, this will be one of the first limits to adding seats.
If money is no object, and we ignore terminal constraints, then no, there isn't. The A380 already supports 850 passengers in a single-class layout, and Airbus has plans to make a stretched version (the wings are designed to give room to stretch the fuselage without needing too much redesign). 1000 passengers isn't really that much of a stretch; we can basically get there now. The issue is that economic obstacles are very real obstacles to building planes; no one has a need for a 1000-person plane on any route (the A380 isn't even used today with max capacity, it's used to give lots of space to first-class passengers with a bit over 500 passengers).
There are certainly at least two airfields out there that would be able to handle a 1000-passenger A380; unfortunately, neither is a commercial airport (the two are Edwards AFB and the Shuttle Landing Facility; the SLF has a 15,000 foot runway that's 300 feet wide, while Edwards has a similarly-sized paved runway). However, if operation is uneconomical, you won't see anyone building a plane to do it.
Aside from finding a route that would make such a large aircraft profitable, the biggest barrier is probably that the existing infrastructure couldn't support such an aircraft. Terminals would have to be enlarged to handle the enormous passage of 1,000 riders. I'd guess that most taxiways probably aren't large enough either. Even the A380 has run into trouble finding suitable airports to operate out of.
NASA sponsored a program to answer just this question. It was called the Ultra Efficient Engine Technology project, or UEET. If you Google it I believe you will find the best estimates for max aircraft size, around the turn of the century when this project concluded, was 2.5 million pounds MTOW. That is a bit more than 2X the size of the A380, and with current tech, that number is probably around 3 million pounds MTOW.
As aircraft get larger, it becomes more difficult to make them strong enough. For example, a two-story building does not move much, but the top of a skyscraper will sway back and forth by about a yard in the wind. Aircraft have the same problem. With super materials, like carbon fiber, it would probably be possible to build an aircraft big enough to hold 1000 passengers, but would require a range of new designs and innovations, so there is a "barrier" in the sense that the builder would have to come up with a whole new set of innovative designs. All of these new designs would then have to be tested, which would be laborious. You may remember the tedious testing process for the 787 "dreamliner" that went on for years, even though it was a relatively standard design. A giant aircraft would have a testing regime that would make the 787's look like a piece of cake.
From a practicality point of view, 1000-passenger planes would not work without changing the gate arrangements at the airport because loading and unloading would take too long. You would need to have 3 gates, 3 doors and 3 ramps all in operation simultaneously to load and unload the aircraft in a time-efficient manner. Since airport terminals are designed to have only one gate per aircraft, you would have to redesign all the airports in the world to accommodate a gigantic aircraft with a triple gate. Starting to see the problems?
Another problem is that runways might not be long enough. The larger the aircraft, the longer the runway must be. A 747-400 has a rejected takeoff distance of around 12000 feet if there is a reverser failure. Usually 747s will not operate on any runway shorter than 7000 feet and that is pushing it. If they have an RTO on a 7000-foot runway, it could result in multiple fatalities. With a giant aircraft, you might need a 12000 foot runway or something like that at least, so it would be difficult to operate the aircraft.
