Not a big deal, if you have some time and real estate.
Assume that you know about noons and solstices, and can catch either by watching a shadow of a post.
During a summer solstice, position two posts along the line of the rays / shadow; the longer the better. Watch the sun hit the same line during a winter solstice. Along the way, notice when, relative to the exact solstice moment, is the noon. Sometimes you'll miss the exact solstice moment because the sun will be below horizon.
A dozen or two years of watching and recording should show you that the positions of the sun during solstices, and the time oof the exact solstice positions relative to the noon, roughly repeat every 4 years.
For bonus points, mark the positions of bright stars, like Sirius, Vega, Arcturus, etc, and notice how they repeat their patterns on solstice days / nights. They will repeat every 365 days approximately (like solstices, yearly), and every 1641 days exactly (4 years).
This all takes a large, flat, undisturbed surface to put posts on, to mark angles.
Not a lot of work, and could fit in one lifetime pretty well, given some prior ideas of watching the sky and measuring angles.
Of course, as you improve your instruments, you will notice how your neat approximations actually are imprecise, etc, and you'll have to invent a Gregorian calendar to spread the error more uniformly :)
Averaging the observations over several years was the insight I was missing. Was trying to understand how to use contemporary timekeeping devices to that kind of resolution in a single cycle.
That's blowing your mind because the currently implicitly or mindlessly acknowledged understanding by the majority of people is that today is the peak of human's evolution. Majority thinks that they inherently and collectively know everything better in relation to what our ancestors knew, especially compared to those thousands of years ago. But how do you really know. Because they had no iPhones and cars? Development of technical tools, yeah there we _might_ well be on the peak, but that doesn't really cover all aspects of actually _understanding_ our earth and the universe.
Keeping a calendar is fairly simple (requires knowing writing). Observing which day the solstice is on is also fairly simple (requires a sundial: the shortest/longest shadow at noon are the solstices. If you're in the tropics just observe when the sun passes directly overhead instead). Observing that the solstice moves by a day every 4 years isn't too hard by analyzing those observations.
The tricky part seems like it would be that one needs precision quite a bit better than one day. So either the measurement can be done over the course of many years or the time of the solstice needs to be determined to excellent precision.
It requires someone to at some point remember that "Hey when I was a kid 20 years ago the solstice was on December 22 but now it's on December 17, what happened?" and then go looking at historical records to generate a trend line with a slope of 0.25 days / year. The measurement doesn't have to be at one point.
The adjustment to 365.25 came during Roman times. The Gregorian adjustment came after well over a thousand years to realize the drift since Julius Caesar's time.
Not to tut-tut too much, because it is interesting the extent to which a lot of science is older than any of us tend to think[1], but this isn't so hard. You need two measurements:
1. A count of days. Have your acyolytes put a rock in a jar every morning, whatever. Tedious, but easy.
2. A measurement of the sun's highest[2] angle each day. Harder, but not that hard. Put a stick in the ground and have the acolytes spend their lunch carefully moving a pebble to see how close the shadow gets to the base of the stick, then carefully paint the ground to show where the pebbles have been. Change the color each year. If it's cloudy, paint a line between the dots to interpolate.
Do this for four years. You'll note that the highest point of the fourth year is 4x365+1 days, give or take. Do it for a decade or two and you'll see that 365.25 isn't quite right either because you're still a little off.
It requires patience and rigor, but those aren't modern inventions.
[1] While a lot of it, including most of medicine, is much younger than we'd expect.
[2] Or lowest. But acolytes prefer to do their tedious stuff in the summer, lest they start engaging in idolatry and moon worship or whatever.
They probably could measure summer solstices, and count the days between 4 of them, and they got 1461. They could've measured 1461 days -43 minutes, and be more precise, if they had wanted. Julius Caesar was as short-sighted as most software developers today, and he was only planning for like 40 years (that's already more than 7 hours off) but not much more.
Internet "influencers" really have us convinced that humans were barbaric apes until a century or two ago and then suddenly invented the modern world. What about all the achievements before then? Could we not have used math and science a few thousand years ago? Nope, it was all aliens of course.
It's only internet influencers if you grew up in the last few years and get most of your world information from them. I'm sure that's more than a roomful of people. Anthropologists have been doing it for well over a hundred years.
There’s a huge contrast between how much human intelligence had achieved scientifically and culturally, and how humanity (at large, on a global scale) behaves itself.
