With regard to the stars, this is possibly the main reason why the ancients (classic Greece) preferred the geocentric to the heliocentric model. They were well aware that the heliocentric model made much more sense in terms of the geometry of the motions described. But the first hurdle was the notion of the Earth in motion, which was possibly just too counterintuitive to be taken seriously. By itself, though, I suspect that wouldn't have been a sufficient objection.
In the Almagest, the classic description of the geocentric heavens, Ptolemy specifically mentions the heliocentric theory but says that in addition to the problem with the Earth being in motion, there's an additional problem. And that's the complete absence of any perceived parallax in the stars over the course of a six-month period (a period which would, in the heliocentric model with the Earth in orbit around the Sun, be the extreme distance traveled from one side of the orbit to the other). This implies that the distance to every star from the Earth at both positions is effectively equal and unchanging; which for their purposes and their notions of distance, implies that the distance is essentially infinite. (For our purposes this is true, too. Just because we learn about distances such as "four light years" and "one hundred light years" doesn't mean that we have any intuitive comprehension of these distances at all. For pretty much everyone, really. The distance from one side of the Milky Way galaxy to the other is approximately 851 quadrillion miles. Doesn't help? Right...because "quadrillion miles" just doesn't work, either. Sorry, there's no way we're going to make sense of these numbers in distances we can comprehend.)
The Greeks thought the stars were attached in some way to the inner surface of a giant sphere...they were referred to as the "fixed stars", as opposed to the stars which moved, i.e., the planets. The notion that the Earth was in orbit around the Sun would greatly simplify the geometry of the whole thing, and that was something that certainly appealed to the Greeks who so loved elegance. Don't think that they were terribly pleased with the epicycles people so often ridicule today (though, really, it's not the epicycles which are most problematic, it's the equants). But placing the Earth somewhere other than at the center of the universe, and putting it into motion, and making the distance to the stars effectively infinite...that was just too much to swallow. They weren't dumb. As Sagan said, extraordinary claims require extraordinary evidence. The Greeks weren't empiricists, but the sentiment is still relevant. We require a lot of convincing to accept something that is far outside our commonsense intuitions. Many things we learn in school and which most of us today accept are, nevertheless, deeply counterintuitive.
This is all with regard to observations with the naked eye. Telescopes can provide images with enough resolution to detect parallax of the nearby stars with respect to those much farther away.