There are, of course, many answers to this question. In any given place and time, individual reactions to a cultural tradition will vary with personal experience, education, and a host of other factors - just as the authority of myth varies from culture to culture and over time.

When I was teaching Greek mythology, I was often asked whether the Greeks really believed that a given myth had happened. In a few cases - say, for the major events of the Trojan War - I could confidently state that most classical Greeks did believe that it had happened, though they disagreed over the details. I could say with equal confidence that I had a good idea what a few elite, educated men (e.g. Plato or Epicurus) thought about the myths. But in the case of most myths, and for the vast majority of ancient Greeks, I couldn't give a good answer.

Greek religion didn't require belief in the myths. The Greek gods, famously, were thought to be much more concerned with what you did for them than with what you thought about them. But the myths were universally known; and although educated Greeks of a philosophical bent were fond of rationalizing them, allegorizing them, or dismissing them as the work of mischievous demons, we have no clear idea what the illiterate majority thought.

Members of elite tended to assume that the masses took the myths literally. One author, for example, observes that gods and heroes often appear in dreams, since most people believe the stories about them. Another claims that the peasants around his villa imagine Agamemnon – the mythical Greek war leader at Troy – to be the current Roman emperor. At least some commoners may have really found aspects of the myths (in particular, the idea that gods were accessible) appealing. There is, however, no way of telling how widespread belief in the myths really was at any given place or time in the Greek world.