Depends entirely on what you mean by "the ancients". Many early civilizations certainly believed the Earth to be flat. In Greece, both Homer and Hesiod in 8th-7th c.-ish BC wrote of the Earth as a flat landmass surrounded by endless ocean, as did the writers of the texts found on coffins and pyramid walls in Egypt. The Indian Vedic lore passed down in oral tradition since the second millennium BC, compared the Earth to a wheel spinning on an axle, although lacked their western neighbor's all-encompassing ocean. In China, the concept of a flat single-landmass Earth persisted as late as the 17th c. AD.

Pythagoras in 6th c. BC Greece is often credited with discovering that the Earth is in fact spherical, although there's some room for debate on if he was building on prior work, sources from the period being few and much of our information coming from later writers. By the time Plato and Aristotle began writing in the 3rd c. BC, the spherical Earth was a given. This appears to be a recent change in overall thinking, as Plato writes in his dialogue Phaedo that in his youth Socrates had sought the answer to the question of whether the Earth was flat or round. "Thus reasoning with myself, I was delighted to think I had found in Anaxagoras a preceptor who would instruct me in the causes of things, agreeably to my own mind, and that he would inform me, first, whether the earth is flat or round, and, when he had informed me, would, moreover, explain the cause and necessity of its being so, arguing on the principle of the better, and showing that it is better for it to be such as it is...".

In the 3rd c. BC, Archimedes began laying the groundwork of proving the sphereicity of the Earth in his treatise On Floating Bodies, in which he noted that a body of water doesn't have a flat surface, but instead "The surface of any fluid at rest is the surface of a sphere whose center is the same as that of the Earth". In 240~ BC, the chief librarian of the Library of Alexandria Eratosthanes was able to to estimate the circumference of the Earth to within a relatively small margin of error, although his exact process has been lost, only a simplified explanation from the 1st c. BC writer (or later, there's some debate based on the details he covers in his On the Circular Motions of the Celestial Bodies) Cleomedes. Whatever his actual methods, Eratosthanes returned a figure of 252,000 stadia, an unfortunately inconsistent unit of measurement, but his margin of error, depending on the value of his stadion, would be accurate to within 1-15%.

Of course, that's not to say that just because the idea caught on in Greece that it was immediately known to be fact everywhere forever and always. In Rome, early Christian writers in the 1st to 3rd centuries AD, such as Methodius, Lactantius, and Augustine, all rejected the sphere view, though this is less relevant to your question about the ancient world.

Of course, this is all only the things known to those philosophers and scientists whose writings exist to this day. We can't say for certain the general knowledge of the average Roman soldier, Greek farmer, or Egyptian mason or whether they were swayed by the teachings of the intellectuals of their times.