Raised in Providence, Barnett studied at the Rhode Island School of Design as a teenager and had his first solo exhibition at Grace Horne Galleries in Boston in 1928. He graduated from the School of the Museum of Fine Art, Boston, and departed to travel through England, France, and Spain. He studied printmaking with Stanley William Hayter in Paris before returning to the United States in the early 1930s.
In the 1930s he split time between New York City and Gloucester, Massachusetts, and exhibited in New York, Philadelphia, and New England. From 1946 to 1951 Barnett was represented by the Mortimer Levitt Gallery on 57th Street in New York. He exhibited in the juried exhibitions at the Pennsylvania Academy of the Fine Arts in Philadelphia and the Corcoran Gallery in Washington, D.C. He taught at Clark University, Yale University's Norfolk Summer School of Art, and the School of the Worcester Art Museum. From 1951 to 1972 he was dean of the Art Academy of Cincinnati. Barnett's landscape paintings are marked by bold, expressive brushwork and a concern for structure.
