The solar system is chaotic in the sense that no matter how well you can measure their positions, their future evolution, at about 20 million years in the future, is not predictable. We will never be able to predict an eclipse that far ahead of time. We won't even know when the seasons are.
You claim that I have a medieval mindset. It seems you espouse a 19th century view of science, of the clockwork universe.
Science doesn't require reproducibility, nor your weaker requirement of "how to reproduce."
Predictability is the key to science, not reproducibility, though of course those are inexorably tied when it's possible to reproduce something. We can predict that fossils of a certain type will only be found in a specific layer of geological strata. We can predict that hurricanes will be created by and affected by certain wind patterns. We can predict that radioactive atoms will spontaneously decay.
Even though we certainly cannot reproduce those.
Chemistry is a field where it's easy to set up very similar conditions to previous experiments ("reproduce") and where the expected confidence of predictability is quite high. Hence my use of it as an example. It's much harder to reproduce an observation of a supernova, but we have pretty high confidence that when we do see one it will follow certain patterns.