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.
Airbus has announced that they have a potential design for a stretch 380 that could accommodate 1000 passengers.
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.