They didn't keep consistent time as we know it. They scheduled things with fuzzy ranges instead of precise timestamps and used public reference clocks which were inaccurate but useful for two people to know the meeting is whenever the church clock says it's 2 o'clock, regardless of the true astronomical time.
Right. And the zenith can be identified by eye or using a sundial to within an accuracy of a minute or so, but not to a matter of a second or two AFAIK. Maybe using a sundial the height of the Eiffel Tower?
It does seem pretty daunting. A tall sundial would not have enough mechanical stability to be accurate, especially due to wind loading. AFAIK at least one of the people who measured the circumference of the earth, did so by observing a shadow at the bottom of a deep well. Still, a shadow would be pretty washed out over the span of, say, 15 meters, unless some sort of focusing optic were involved.
A tall cliff? A building with a spire on top? An obelisk? Guess now I'm going to have to go and find a tall building with a spire and check how recognizable the shadow of that spire is!
I'd certainly be interested in knowing. One thing that fascinates me is how the ancients made any headway with tools that were so crude compared to what we now have. For instance I always enjoy reading a non-algebra or non-calculus based math proof for something where I would instinctively turn to one of those tools.
One thing it shows is the irrepressible drive of human curiosity.