He showed his work at the Alexandre Gallery in New York and his work is part of the Smithsonian National Gallery of Art. He was a member of the National Academy of Design, The Century Association, and the American Academy of Arts and Letters. He received numerous awards through his life and has been included in many books, catalogs, and magazines.
Barnet studied at the School of Museum of Fine Arts in Boston from 1927 to 1930. He was the official printer for the Arts Students League in New York starting in 1934 where he also taught painting. He lived and worked in New York where he died in 2012 at age 101.