Jessup was born on August 12, 1877, in Richmond, Indiana, the only child of Albert Smiley Jessup and Anna (Goodrich) Jessup to survive infancy. His paternal forebears had come from Virginia before the War of 1812 as part of the great Quaker migration from the Carolinas and Virginia that made the Whitewater Valley of Indiana the "Jerusalem of Quakerism for all the Northwest. " His grandfather, Levi Jessup, helped manage the Friends Boarding School which became Earlham College in 1859 and which his father attended. His mother, not a Quaker, came of a family that included merchants and politicians; a cousin, James Putnam Goodrich, served as governor of Indiana, 1917-1921.
Jessup's mother died when he was eleven. In 1890 his father married Gulia E. (Hunnicutt) Jones, a teacher and the widow of a Friends minister, who bore him a son and a daughter.