The song itself doesn't talk about a setting, but since it set the scene for Saturday Night Fever, a 1977 film about Brooklyn's disco scene, we'll take a look at that.
Saturday Night Fever follows Tony Manero, the king of the Brooklyn disco scene, who is played by John Travolta. Manero holds a dead-end job in a hardware store; he and his three best friends have no prospects and little ambition. But Manero is a slick dancer who spends all his money on nightlife polyester, and, every weekend, he finds a few hours of meaning and triumph at the disco.
The film was based on a supposedly factual account of New York disco culture written by Nik Cohn for New York Magazine called "Tribal Rites of the New Saturday Night." According to Cohn's piece, discos were filled with young working class people aged 16 through 20 for whom the 1960s meant nothing. Instead, their values harkened back to the 1950s and the "golden age of Saturday nights." For this generation, times were too hard to indulge in poetry and protest. They were forced to take any job they could find, work hard, and enjoy what little pleasure they could on Saturday night.
The central figure in Cohn's article was Vincent, who was an Italian-American who worked in a hardware store, just like Tony Manero in the film that came later. Vincent was "the best dancer in Bay Ridge—the ultimate face." But he was oppressed by the awareness that he was growing old. At eighteen and a half, he would soon be too old for the scene. "By natural law someone new would arise to replace him. Then everything would be over," Cohn wrote. The article was compelling and the film it inspired was generally well received.
Alas, Cohn later admitted that much of the piece was fabricated. A Brit who was baffled by Brooklyn's subcultures, Cohn had actually based Vincent on a London mod who'd been around in the 1960s. However, even Cohn admitting that he'd made up most of his article didn't undo the cultural legacy of Saturday Night Fever.