I think it's easier to just sort of think of this like probability and statistics.
I start with an assumption: the universe does exist, it's real, and humans can perceive the real universe, although likely with a bunch of filters and biases. Then I also assume that through careful work (identifying filters and biases, setting up good processes to avoid them) we can see the real universe in a truthful way (Plato's cave allegory, and Descartes' Great Deceiver are both interesting explorations into the limits of this idea).
I figure if either of those assumptions is violated (for example if there is some sort of Great Deceiver) we can't really supercede them.
Then, based on those assumptions, I mainly focus on empiricism combined with high quality experimental technique, and a relentless drive to reduce variables which don't contribute directly to the system under study. Think of Galileo's experiment dropping two balls of different mass. When he conducted it in air, if he had used a feather, many people would conclude that heavier objects fall faster. But if air effects are excluded (in vacuum), we can see that feathers fall at the same velocity as heavy balls. This requires some leaps of intuition and also some very careful experimental setup.
Once you've got to that point, you're well set up to start using the great tools of science: microscope, telescope, spectroscope, precision clock. Those immediately give you access to a whole collection of additional high quality data that is not directly available to our senses and intuition. They allow you to collect additional data, and help make confusing things like diseases and stars make a lot more sense (germ theory of disease, our sun is a star that is just very close, and our sun is composed of elements that exist on earth).
Much of what science then studies is as close to objective as we can realistically get- I don't worry too much that gravity, space, time, etc are significantly different from how we perceive them in 4d (it would be interesting if our mental filters really do warp the real universe in a way that we see a very false representation of it).
Once you've reached that point you can build satellites and reduce the likelihood a flat earth compared to a oblate spheroid model. But then it starts to get hard: you will discover relativity and quantum mechanics, both of which are not intuitive and in fact completely counter all our normal expectations. yet they are key theories that help us make and improve technologies like GPS and transistors!
Right now some of the most interesting questions in science are about things we don't have any good experimental apparatus to probe. QM is a great example- we can do all sorts of amazing things with QM in the lab, but our brains can't wrap around the underlying physics at all. Is QM the theory wrong? Is there a better theory, one that's more intuitive and consistent with our daily lives? Probably not; as much as I like the idea of a deterministic universe, it's just an unlikely hypothesis to test.
I don't think post-modernism really was the origin of the idea that objectivity was an ideal, not a real. I think most empiricists have implicitly assumed that we have to work really hard to not fool ourselves.
I once saw an interesting talk, where the presenter didn't understand the difference between relativism and relativity. So they argued that their cosmology (native american) was just as valid as Einstein's theories. I can't really see that as true. I think under any reasonable subjective determination, empirically determined theories of cosmology will do a better job of explaining the observed universe than a spiritually derived one.
I recently learned of a term, "scientism" (after being accused of it while arguing with a religion and philosophy professor who likes to mix buddhism and christianity).