From a purely aerotechnical standpoint, it's not a problem. Short-space recovery of aircraft can be done in any number of ways. Some aircraft have such a low stall speed that they can take off and land in a matter of feet. In the video below, you can see a plane take off in just 15 feet and land in the space of 10 feet. That is just about the overall length of the plane itself, so the aircraft needs essentially maybe 20 feet of runway. And that's a relatively modern plane that is at least capable of carrying a person. By the time it lands, it's no faster than a walking pace.
It depends somewhat on what kind of performance you want your planes to have, but there is rarely only one solution to a technical problem. We use long, wide open runways because we want fast, heavy planes that are easy, safe, and more comfortable to land, but that doesn't mean that runways are the only way to launch and recover planes.
Launching and recovering heavy, fast planes could be a bigger challenge, but still solvable.
The very first plane, the Wright Flyer, was launched with a catapult. The engine was underpowered- so it could sustain flight but the aircraft could not take off by itself. In early attempts they could use the wind to get the Flyer airborne, but wind was not a reliable method. The catapult was a small tower with heavy weights and a pulley and rope system- when released, the weights would pull the Flyer about 50 feet along a rail at which the pilot nosed up the plane and would become airborne. This is a neat approach for your world, because you're trading horizontal space for vertical space by using a right-angle pulley. Modern aircraft carriers use catapults driven by steam and electric motors rather than dropping counterweights.
Landing a fast/heavy plane can be done in different ways. Others have already mentioned tailhook systems used on aircraft carriers. Those systems also have a crash net that can be used in the event the tailhook fails (but nets + propellers or jet engines is a bad time). Entire aircraft have been landed with parachutes. They have been snagged out of the air by moving arms. They have been stopped on short runways by using retrorockets.