The drawing below depicts the structure of the Theater of Dionysus in Athens. The action occurs in the orchestra; a platform were the actors stood during performances. Since the Dionysian festival was religious in nature, an altar was located in the center of the orchestra for sacrifices.
The Skene (original meaning “tent”) was a large wooden wall used for the backdrop. It could also be decorated as part of the set. The Skene was a substantial structure because it had to incorporate a lifting apparatus to be used to suspend actors in the air. In the case of one play, the entire chorus went to visit Zeus and “flew” to Mount Olympus. There were two (or three) openings cut in the Skene connected to long ramps called Parodos. These ramps were used for entrances and exits for the actors.
The Paraskenia was a roofed building which housed the dressing rooms and costumes for the actors. Next to the public seating was a roofed building called the Odeon of Pericles which provided shelter in case of inclement weather.
I looked at the many pictures of the Theater on the internet and few if any depict it as it was during the time of Pericles. Some show a Skene constructed out of stone, others a stage. These either represent later versions of the original theater or a reconstruction under the period of the Roman occupation.