Just a clarification: going to space per se isn't what makes you weightless. Gravity affects objects on the ISS (about 400 km above Earth's surface) almost as much as it would on Earth. The key difference is being in orbit, which is equivalent to free fall. Your feet don't exert a pressure on the floor (and vice-versa) when both you and the floor are in a free fall.

Things are different when you go deeper into space, far away from any planet or star, where the actual strength of gravity drops off inversely proportional to distance squared.

Jules Verne's "From the Earth to the Moon" includes a scene where the characters experience weightlessness, but only at the point where the earth and the moon's gravity cancel each other out. This is a step in the correct direction but obviously not a full understanding of what it would be like in space.

I'm not exactly sure when the tipping point came, but by the 1920s weightlessness was pretty well understood. Herman Noordung's 1929 book The Problems of Space Travel - The Rocket Motor discusses the physiological and practical effects of weightlessness in pretty minute detail, even getting into topics like writing in space. Fritz Lang's "Frau im Mond" (which had Hans Oberth as a technical consultant) featured a spaceship with foot straps on the floor so that the characters could "walk" in zero gravity, so it was even a part of popular culture at this point.

It was still an open question at that point if a human could survive more than a few seconds of weightlessness, and by the early 1950s both the US and USSR were conducting sub orbital rocket flights with animals in order to gain practical experience in this area. The question of if a mammal could survive more than a few minutes of weightlessness was settled in 1957 when the Soviets put Laika into orbit, however there was still some nervousness about what disorienting effects it would have on humans. When they sent Gagarin up the controls for the ship had a plastic cover on them that could only be removed by entering a special code.

So, yeah, by 1951 it was pretty well established that spaceflight would involve weightlessness. However, it wasn't well understood if humans could survive in such an environment. One idea that was proposed (and is still possible though it's never actually been done) would be to have a section of the ship that would rotate in order to provide artificial gravity for the crew. A well informed author might have figured that such a system would have been a necessary component of any future spacecraft.

Hope this was helpful. Some good books to read on the subject would be Mary Roach's Packing For Mars, and Noordung's The Problems of Space Travel which is actually available for free online from NASA.