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 :)
(Looks at the news) Sadly, still pretty much are.
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.
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.