Barnum and Bailey was an entertainment attraction on Earth in the 19th and 20th century. P.T. Barnum was one of the founders of this circus.
This circus was mentioned in the composition "It's Only a Paper Moon". In the song, the singer spoke of himself and his lover living in a Barnum and Bailey world, which was based on both phoniness and make-believe. (DS9: "It's Only a Paper Moon")