What it was: An early Soviet manned lunar mission plan, arguably the first. Three elements would be launched into Earth orbit, docked with one another, and then launched on a figure-8 lunar flyby and return.
Details: The Russians were planning for a manned Moon mission from very early on. In 1959 they studied using a Vostok capsule to send one man on a loop around the Moon, but realized that it wasn’t feasible. The Vostok could only hold one person, which was problematic for a six-day mission, and its round shape meant that its ballistic return from the Moon (which implies an 11 km/s re-entry speed instead of the 7 km/s of low Earth orbit) would be hard to handle. It would also have to return somewhere near the equator rather than on land in the Soviet Union as preferred.
As a result, Sergei Korolev’s OKB-1 got to work on a new capsule with two cabins. One would be cylindrical and used to house three cosmonauts, while the other would be used to return to Earth. This second cabin was acorn-shaped, which allowed a skip trajectory: it would come in over the Indian Ocean and bounce once off the atmosphere, bleeding off speed and pushing the craft north over the Kazakh SSR. Taken together these two components were dubbed the Soyuz A, also known as the 7K, and it was the ancestor of the very successful Soyuz capsule still used today.
From the standpoint of alternative space races, it was the rest of the plan proposed in March 1962 that catches the eye. As well as the Soyuz A, two other components were planned at the same time, the whole making up a Lunar flyby craft.
The Soyuz B/9K was a rocket block. It would be launched unfuelled to save weight, and placed in Earth orbit. Then another launch would send up a Soyuz V/11K, which was a tanker. Remotely controlled from the ground, the Soyuz B would dock with the Soyuz V and receive its fuel load. The tanker would then be cut loose. Three launches of three Soyuz Vs would fill up the rocket block completely, at which point the Soyuz A and its two astronauts would be launched to dock with the ready-to-go Soyuz B. Once mated, the combined Soyuz A/B would ignite its engine for a journey around the moon and back.
The plan was approved by Soviet leadership on December 3, 1963. Construction of the craft began a few weeks later, with first test flights scheduled for late 1964.
What happened to make it fail: Sergei Korolev became concerned that five launches and four docking maneuvers in orbit would be too hard to pull off. In 1962 he decided that a better approach would be a Lunar Orbit Rendezvous (LOR), which would reduce the weight of the spacecraft to the point that it could be achieved with one launch with a larger rocket—not only for a flyby mission but a lander mission too. This was the same decision reached by the Apollo program.
At the time, these two missions were to be a follow-up to the Soyuz Complex missions, but that eventually changed. Political infighting with Vladimir Chelomei had the Soyuz A/B temporarily replaced with his LK-1 in mid-1964. Nikita Khrushchev fell not long after and, as he was Chelomei’s ally, Korolev managed to get the lunar orbiter and lunar lander mission returned to him a year later. By then there was no time for the Soyuz Complex flybys as well as the later LOR flyby and landing. To make up for the lead the Americans now had, the former was jettisoned from the program and efforts switched to getting the latter two done more quickly. Soyuz A would be used in the single-launch plans, but the other two modules were cancelled altogether.
What was necessary for it to succeed: Korolev coming up with a way to keep Khrushchev off his back.
In retrospect, the Soyuz Complex may have been the Soviet Union’s best bet to beat the Americans to the Moon. All other Moon missions they came up with required either a Proton rocket (which had growing pains until 1970 or so) or an N1 (which had the same but squared); this mission called for the tried and true R-7 derived Soyuz launcher that had been going up since Sputnik. The Soyuz capsule was made relatively safe quite quickly too, so all the eventually-built components that the Complex used have been proven workable. Whether the same is true for the hypothetical 9K and 11K modules is another question, but they were not particularly byzantine in their designs so prospects there were good too.
Against the relative simplicity of the mission hardware we have to place the complexity of the mission profile. While an Earth Orbit Rendezvous is admittedly more involved than a Lunar Orbit Rendezvous, the Soyuz Complex had two more things going in its favour. First, automated docking turned out to be a Soviet strength: they managed to perform history’s first docking between two spacecraft in October 1967 and another in April 1968. They then did so with manned Soyuz capsules in 1969. If anyone was going to pull off the necessary dockings, it was them. At worst they would have had to make several attempts at doing it, but the mission profile was unlikely to kill any cosmonauts so they could have just kept at it until one series of dockings worked out.
The available slack time is the 7K/9K/11K’s other big advantage: it had time to get its bugs worked out before Apollo, or at the least before the post-Moon landing letdown in funds and interest. Its plan was finalized in December 1962, at which point in time the American program was still six months away from finishing Project Mercury. While NASA did at least consider using a Gemini for a Moon mission, they decided to focus on Apollo and getting three men to Lunar orbit.
This opened a window for the Russians as they were content with the two that the lighter Soyuz A would carry, but they squandered it with the bickering between Korolev and Chelomei that saw Khrushchev back the latter until his fall. Korolev and OKB-1 ended up not having a finalized plan until the middle of 1965, and by then Korolev had made the justifiable mistake of switching to a single-launch, no-docking profile that left him at the mercy of Valentin Glushko’s Proton and that rocket’s initial problems.
On top of that Korolev’s OKB-1 design bureau, which constantly verged on oversubscription, was distracted by Khrushchev’s insistence on an earlier multi-man craft. They ended up having to work on the dead-end Voskhod craft in 1964 and 1965. As a result the design, construction, and testing of the Soyuz capsule was heavily delayed; the first successful Soyuz flight wasn’t until October 1968. By then it was all over except for the shouting.