Sailors have been putting to sea in boats and ships for thousands of years, but it was only in the 14-1500s that really long voyages started to become more common place as trade grew and countries looked to expand their territory. For centuries before that, inspite of some of the great early scientists making observations that suggested the world was spherical, most mariners and probably as a consequence most ordinary people believed the world was flat. It looked flat so why would they be interested in thinking anything else? They were more interested in finding enough to eat, paying their taxes and avoiding going to hell by obeying the church. Every so often, a ship would fail to return and word would spread that they had fallen off the edge of the world.
As time moved on, some of the work of ancient mathematicians and astronomers started to become useful to mariners and navigators as they were able to start using known positions of stars, the moon the sun and the times that these celestial bodies were predicted to be in certain positions. Using rudimentary instruments, the navigators were able to get a better idea of where they were heading, how far it was and were able to find places noted by other mariners. The navigation was subject to the unpredictability of weather, currents, and the difficulty of taking accurate sightings with primitive equipment, but was enough for those who had some knowledge in the matter to start making rudimentary maps of their expeditions for others to follow.
Another interesting development was that it was discovered pretty early on, when sailors started sailing well offshore, that by putting a lookout 50 ft up the mast, they could site land much earlier than they could from the deck. This led to the 'crows nest' which was the lookout position on all seagoing ships right up to the invention of radar. This was also backed up with placing land based beacons on the highest points of the coast as navigation aids, allowing mariners to see them at far greater distances than if they were at sea level. The revolution over a few hundred years in navigation at sea, was largely responsible for the change in opinion from the earth being flat to being spherical. The physical observations of mariners was backing up the findings of the mathematicians and astronomers.
Of course there were always occasional false readings caused by temperature inversions and weather conditions reflecting images from further away as in a mirage in the desert. But as understanding of these improved together with a steady development of the equipment, navigation equally became more accurate. I would add that as an offshore sailor for many years, I have frequently navigated accurately to distant shores and also observed the difference that height makes to observations at sea.
I seriously doubt that there are any offshore sailors that are flat earth believers, but would be interested to hear the views of FEs on my comments.
Roger