The Greeks were perhaps the most advanced in terms of cosmology of all ancient cultures. We’re not really sure if the ANE cultures actually believed the universe to be shaped like a snowdome, but we do know the Greeks believed the universe was comprised of the Earth in the center of giant spheres.
Although there is no material evidence of much of the work done by Greek philosophers between 600-300 BC, it is believed that Anaximander (c. 610 BC–c. 546 BC) described a cyclical Earth suspended in the center of the cosmos surrounded by rings of fire, and that Philolaus (c. 480 BC–c. 405 BC) the Pythagorean described a cosmos in which the stars and ten bodies including the planets, the Sun, the Moon, the Earth, and counter-Earth (Antichthon) circle an unseen central fire.
Looking at the night sky the ancient Greeks found two primary kinds of celestial objects; the fixed stars and the wandering stars. Think of the night’s sky. Most of the visible objects appear to move at exactly the same speed and present themselves in exactly the same arrangement night after night. These are the fixed stars. They appear to move all together. Aside from these were a set of nine objects that behaved differently, the moon, the sun and the planets Mercury, Venus, Mars, Saturn and Jupiter each moved according to a different system. For the Greeks these were the wandering stars.
In this system the entire universe was part of a great sphere. This sphere was split into two sections, an outer celestial realm and an inner terrestrial one. The dividing line between the two was the orbit of the moon. While the earth was a place of transition and flux, the heavens were unchanging. Aristotle posited that there was a fifth substance, the quintessence, that was what the heavens were made of, and that the heavens were a place of perfect spherical motion.
By the time of Plato and Aristotle, the consensus view among educated Greeks was the Two-Sphere Model of the universe (as Thomas Kuhn later called it): A spherical earth, fixed at the center of the universe, surrounded by an enormous celestial sphere, holding the stars and spinning around us once a day. The sun, moon, and planets were presumably somewhere in between, carried around in their circles by similar mechanisms.
Around the 4th to 3rd centuries BCE the Greeks, under the influence of Aristotle who argued that the heavens must be perfect and that a sphere was the perfect geometrical figure, exchanged this for a spherical Earth surrounded by solid spheres. This became the dominant model in the Classical and Medieval world-view, and even when Copernicus placed the Sun at the centre of the system he included an outer sphere that held the stars (and by having the earth rotate daily on its axis it allowed the firmament to be completely stationary).
We might scoff at their “naive” beliefs, but based on their observations, this model of the universe could not be refuted until thousands of years later with advanced methods of measurement.