They didn't believe in planets as we know them, they believed in an Earth centric system where the moon and sun revolved around us, and as far as an infinite plane they believed that the horizon was the end of the Earth.

There is a popular myth about Columbus proving the world is round, most societies already knew this, he believed the Earth was smaller than it actually was and since most societies in Europe had no idea about the New World they thought he would die in the middle if the Ocean from lack of food, they turned out to be right about the size of the Earth and had the New World not been there he would have died on the Ocean.