Born in 1911 in Beverly, Massachusetts, Barnet knew by the age of ten that he wanted to be an artist. As a student he studied with Philip Leslie Hale at the School of the Museum of Fine Arts, Boston and viewed first-hand John Singer Sargent at work on the murals of the Boston Public Library. In 1930 Barnet studied at the Art Students League of New York, with Stuart Davis and Charles Locke, beginning his long association with the school.