Barr was born in New York City in 1950. His father, Donald Barr, taught English literature at Columbia University before becoming headmaster of the Dalton School in Manhattan and later the Hackley School in Tarrytown, New York, both members of the Ivy Preparatory School League. Barr's mother, Mary Margaret (née Ahern), also taught at Columbia.[12] Barr's father was Jewish and raised in Judaism but later converted to Christianity and joined the Catholic Church. His mother is of Irish ancestry. Barr was raised as a Catholic.[13][14] Barr was the second of four sons, and his younger brother Stephen Barr is a professor of physics at the University of Delaware.[15]
Barr grew up on New York City's Upper West Side. As a child, he attended a Catholic grammar school, Corpus Christi School, and then the non-sectarian Horace Mann School. After high school, he attended Columbia University, where he majored in government and graduated with a Bachelor of Arts in 1971. Barr was also an active member in the Sigma Nu fraternity. He did two additional years of graduate study at Columbia, receiving a Master of Arts in government and Chinese studies in 1973. While at Columbia, Barr opposed anti-Vietnam War occupation protests by students on campus.[16]
After moving to Washington, D.C., to work as an intelligence analyst for the Central Intelligence Agency (CIA), Barr entered the evening student program at George Washington University Law School. He graduated in 1977 with a Juris Doctor with highest honors.[17][18]
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