The quality of work, done by hand, is just staggering when you think of the consequences that could have transpired if mistakes had been made.
See e.g. https://ntrs.nasa.gov/search.jsp?R=19680020624 for Gemini.
The movie also shows Dorothy Vaughan teaching themselves how to program those first machines. Remember that pretty much noone had any programming experience, and there are still pitfalls today for people doing science calculations on a computer.
> As a computer, she calculated the trajectory for Alan Shepard, the first American in space. Even after NASA began using electronic computers, John Glenn requested that she personally recheck the calculations made by the new electronic computers before his flight aboard Friendship 7 – the mission on which he became the first American to orbit the Earth.
[1]: https://www.nasa.gov/feature/katherine-johnson-the-girl-who-...
They were large boxes with nixie tubes, the size of a modern midi-tower PC, only horizontal. They were limited to relatively simple operations, like square root or ln. They were more precise than a slide rule, though.