The short version: the U.S. military once had a rocket-propelled, near-vertical landing and VERY short take off
In 1980, the U.S. military was in a peculiar position. Its vaunted forces had been apparently rendered helpless by the Iranian seizure of 52 American hostages from the United States Embassy in Tehran. The first military rescue attempt, Operation Eagle Claw, ended in a highly visible failure when two aircraft collided at the rendezvous point Desert One - the name by which the botched operation would be remembered. Another solution was needed, and quickly.
Special Operations Forces planners decided that the primary weakness of most of the plans submitted for consideration was their reliance on helicopters. Desert One had been necessary in part to refuel the short-ranged machines, and other plans involved either more of the same or even airlifting, assembling and then abandoning light helicopters. Desert One had also demonstrated the difficulties of flying the relatively more complex machines over and through desert sands. After the failed attempt, the hostages were split up and moved to various locations away from the U.S. Embassy in any case.
However, one audacious bit of planning decided that while helicopters weren't viable, air transport was required. Close to the Embassy was the only available open space - directly across the street was Amjadien Stadum, a soccer (er, sorry, football) arena with a standard 100-meter field, surrounded by bleachers. This had been identified as a rally point for the helicopter-based missions, as it was a visible and easily-accessible spot for helicopters to land and take off from. The planning group was surveying all available options in the U.S. arsenal, and wondered if it was possible to get a fixed-wing aircraft into (and out of!) that spot.
The U.S. Special Operations Forces already had something close - the MC-130 Combat Talon transport. Designed for short and rough field operation, it nevertheless was usually operated from 3,000 feet of runway. Could it be modified to provide the capabilities needed?
Lockheed (the manufacturer) and the U.S. Air Force (the operator) both thought so. A crash project was initiated under the auspices of the overarching Project Honey Badger (no, I'm not kidding, that was the name) to modify several MC-130 aircraft for the role. The project, named Credible Sport (no doubt a nod to the target landing zone) took two proven technologies - the MC-130 and standard JATO bottles - and combined them into a frankenbird. The XFC-130H Credible Sport was an MC-130 with no fewer than thirty rocket engines mounted to it. The concept called for first modifying an MC-130 to give it as slow and steep a descent speed as possible; the final prototype was able (through the addition of dorsal end ventral fins for stability, Fowler flaps and larger ailerons, among other changes) to descend at just 85 KIAS on final approach using an eight-degree glideslope - more than double the standard 3 degrees used in 'normal' aircraft operations. Once that was complete, they called in experts from Naval Air Weapons Station China Lake to help essentially stick rockets on the thing in every direction.
No, seriously. There were two sets facing directly forward, one pair mounted at the top of the fuselage and one pair mounted near the bottom. Another set faced directly down on either side of the aircraft. These would allow it to land; when the airplane was just over the landing area (actually, just before it) it would fire the first (upper) set of forward-facing rockets, which would decelerate it sharply. Almost at the same time, it would fire the downward-facing ones to counter the increased sink rate that the sudden deceleration would cause, and as it touched down, it would fire the lower forward pair to brake it to a stop on the ground.
Once ready to depart, several sets mounted on the rear of the fuselage at a 45 degree downward/aft angle would fire, more like normal JATO, to lift the airplane off the ground and help accelerate it to flying speed. A set of smaller rockets extended on pylons would also fire to keep the aircraft balanced and pointed straight during transition to normal flight, and a pair of rockets on the tail would fire to prevent the tail from striking the ground during takeoff. These weren't standard aircraft rockets - the forward-facing ones were taken from the ASROC system (four sets of two); the vertical-descent rockets were eight motors taken from AGM-45 Shrike missiles. The rear-facing ones were taken from the RIM-66 Standard shipboard missile. The two pylon mounted were also Shrike rockets, and the tail rockets were more of the Standards.
That wasn't the extent of the modifications, however. In order to ensure that any wounded evacuees would be able to receive care as soon as possible, the mission plan called for the airplane to land on an aircraft carrier nearby. As a consequence, an arrestor system (read: big ol' tailhook) was added to the airplane as well, and crews spent over 400 hours practicing the use of this system in various engineering simulators. To assist in navigation over desert, a very early GPS navigation system was added. Terrain-following radar, to allow low flight. In order to permit the crew to land without lights, a FLIR turret was added; it had to be specially mounted ahead of the nose in order to not be blocked by the nose gear during landing.
The thing of it was - it worked. The first complete aircraft, Air Force number 74–1683, was delivered to Wagner Field - a then-unused part of Eglin Air Force base, once used by the Doolittle Raid pilots training for their famous raid - in Florida for secret testing. It was flown in various configurations, everything from single-rocket tests to final full-up test operation. On landing, it could stop in less than 200 feet from touchdown. On takeoff, it left the ground in less than 150 feet, and at 300 (the length of the soccer field) was over 50 feet up, able to clear the surrounding bleachers. However, the project was in the end a failure. The rocket firings were controlled by a computer, but there were manual controls in case the computer failed and the aircraft crews were trained to operate them in manual mode. During the final test landing the crew was operating the systems manually. The upper forward rockets fired normally, but the downward-facing rockets did not fire - and the flight engineer, confused and blinded by the enormous burst of rocket bloom outside the cockpit - misinterpreted the jolts and thought they were already on the surface. As a consequence, he fired the lower pair of braking rockets while the airplane was still some twenty or thirty feet in the air. That, coupled with the fact that the verticals did not fire, caused the aircraft to essentially drop straight down onto the pavement, at which point the right wing broke in half between the two right wing engines (three and four). The fuel tanks ruptured, and a huge fireball engulfed the right side of the airplane. There is footage of these tests widely available on YouTube - search for 'CREDIBLE SPORT' or 'YMC-130' and you will have no trouble finding it. It's impressive.
Fortunately, since it was a test, emergency personnel were close by - within eight to ten seconds, the first had reached the airplane. The fire was extinguished, and the crew exited the airplane without injury. No one was harmed. The airplane, however, was a loss - and to maintain secrecy, it was first stripped of its new systems and then destroyed and buried.
The other two airplanes in the program - one complete, one which was used as an individual system testbed - did not fly in anger. Before the problems could be corrected, the Algerian negotiations (and, some say, the election of Ronald Reagan to the presidency) resulted in the release of the 52 hostages. One of the two surviving airplanes, number 74-2065, was stripped of its new gear and sent back into regular use.
Over the next year, Operation Credible Sport II was tasked with evaluating the various systems to determine if they could be built for standard use. The third airplane from the original program, serial 74-1686, was sent to Robins Air Force Base in Georgia, where it was redesignated YMC-130H and used for this evaluation. The eventual decision of the project was that the V/STOL C-130 could not be made safe enough for peacetime operation risk management practices, and the project recommended instead that the non-rocket modifications to the aircraft be adopted for Special Ops MC-130s (the lower approach speed and increased glide slope). Those were rolled into the mainstream MC-130 fleet with the production of the MC-130 Combat Talon II. The test airplane, 74-1686, ended up being decommissioned when it was determined that removing the full set of unused systems left over from Credible Sport I and II would cost more than it was worth. In 1988, it went on display at the Museum of Aviation at Robins Air Force Base, where it finished its military career. The last airplane, 74-2065, continues to fly today with the U.S. Air Force, its systems returned to 'normal' (or whatever that is for a special operations airplane).