The unusual shape of the piazza isn't a baroque affectation but precisely follows the ancient perimeter of the Stadium of Domitian that once stood on this spot. The stadium was a grand edifice ordered by the emperor in the first century BCE; it had a rectangular shape with rounded short sides, was completely covered in white marble and could hold up to 30,000 spectators!
As opposed to other Roman structures built to house spectacles, such as Circus Maximus or the Colosseum, the stadium of Domitian didn't offer chariot races or gladiator fights: here were played the "Agones", the games in honor of Jove, and the athletes arrived directly from Greece for the sporting matches where even little girls challenged each other in races. The name of the piazza is derived precisely from the Greek term "Agone"--contest--which in the Middle Ages became "in Agone" and finally "Navona".
As in almost all the places destined for fun and games in ancient Rome, the stadium was surrounded by taverns and lupanari (ancient houses of ill repute). The prostitutes would snare customers by exhibiting paintings illustrating their specialties. Female slaves, on the other hand, were shown naked to possible buyers.
This was the fate of Sant'Agnese, a Christian virgin: legend has it that she was protected from indiscrete eyes thanks to the sudden miraculous growth of her hair and it is to her and the spot where she was martyred that the church designed by Borromini stands.