Flesch was an Austrian born in Vienna. He earned a doctorate in law from the University of Vienna in 1933. He fled to the United States to avoid the imminent Nazi invasion and antisemitism. In the United States, Flesch became a graduate student at Columbia University, where he earned a PhD in Library Science. He also met Elizabeth Terpenning, whom he married. They had six children, five daughters and one son. Flesch lived for most of his life with his wife and children in Dobbs Ferry, New York, a village in southern Westchester county.