There is a double planet system. Both planets are habitable. One planet has humans and the other one animals. So instead of seeing a grey Moon in the sky, the humans living on the first planet see another world with forests, oceans, rivers, etc. So it is widespread knowledge that the "Terra" is just another planet, that humans live on a sphere. I want these double planets to be closer to each other than our Moon is to our Earth, but I want to have a scientific facts whether that is possible or not. For long time, humans on planet 1 (the "Terra") have looked at the habitable planet 2 (the "Luna"), but they haven't been able to reach it yet, until now.
On the "Earth" there is a civilization with pre-modern technology. By that I mean that they haven't had the industrial revolution yet. But they are slowly going into an industrial evolution. I imagine this civilization being similar to medieval China. Medieval China was known for having a strong totalitarian government, and many advanced mechanical inventions. The Roman empire would also be a good example describing this one empire on the planet "Terra".
The most advanced civilization on "Terra" is an empire with a strong government, prosperous economy, and a current level of technology similar to Leonardo DaVinci, medieval China, or Hellenistic Egypt. This empire is starting to undergo an industrial evolution, with the steam engine being just invented. The government heavily finances science and technology, employing geniuses similar to Archimedes or Newton. So they have access to both mechanical engineering knowledge, and mathematics physics such as calculus. Although cars and trains have not been invented yet and all travel and work has to be done by beasts of burden. The empire is about the size of maybe 2/3 the United States, with lots of fertile land, natural and man made waterways, coal and metal deposits near the surface. It is ruled by an emperor as a supreme monarch with a centralized bureaucracy similar to medieval China. The empire is now currently stronger than it has been ever before, with it's major rivals defeated, and no internal or external threats.
The totalitarian government of this empire can make it's citizens do whatever it pleases. The empire has built pyramids, great walls, canals, bridges, aqueducts, and roads without too much difficulty, and zero social protests. Now they choose to go the the double planet "Luna". The emperor gave his word, to employ all the resources of the entire empire for achieving the singular goal of landing people on the "Luna" as representatives of the empire.
Assuming that the atmosphere and the biosphere of the "Luna" is compatible with "Terran" life, including humans, who do not need a special suit or anything to live there. "Terran" animals and plants can survive in "Lunan" environment, and vice versa. Sure the animals and plants of both these double planets have different evolutionary histories, but they still have the same DNA cell based biological structure. Perhaps they were descended from the same microbes which seeded both these double planets.
Just like medieval China, the empire has knowledge of rockets, having used them successfully in warfare to conquer less advanced civilizations. The plan is to somehow be able to get people, animals, and plants from the empire up to the "Luna", have them survive the landing (or use parachutes which are known by then), then establish a colony for the empire, and after some period of time march to a certain region, and light up a huge fire that could be seen from the "Terra" as a sign of success. Because the "Luna" is easily seen from the "Terra", maps of the landscape have been made in advance. It is a one way trip only, and other than lighting up large fires there is no method of communication. The empire doesn't have electromagnetic communications technologies yet.
Assume that the government is determined to get humans up to the other planet, using all the economic, human, and technological resources that they have available, and can try to extend their resources by encouraging progress in the related fields. Assume that there are little to no inefficiencies in the government, no social unrest, and the people can be brainwashed to do almost absolutely anything. Also assume that this is a long term goal. It's not a space race. The empire can work on this for hundreds of years, gradually advancing their technology through the industrial evolution. But they will not give up until they can see the outlines of towns on the other planet. Either the emperor is extremely long lived, or the entire dynasty has the same mindset.
Could such an empire realistically achieve such a task? If so, how would they do this? I mean, what would be the absolute minimum level of technology to establish a colony on the "Luna". If they need to wait hundreds of years until they develop cold war era rockets, they will do just that. But can they get to the "Luna" before reaching that level of technology though? Can they do it using (medieval China style) solid fuel rockets alone? Can we think of any other alternative launching systems that can be made with pre-industrial or early industrial technology? And what about landing on that other planet? And they also need to lift not just two men, but a whole population of men, women, farm animals, and plants in order to establish a colony on the first try. There is no way of going back for the colonists.
If there is a way to modify the planets to make this story more plausible, such as decrease the gravity, or make the planets closer together, that is acceptable, but it has to be science physics based.