Barnet was born in Beverly, Massachusetts, the youngest child of a large Eastern European family. During his early upbringing in the small coastal town, he forged a deep connection to the landscape, architecture, and atmosphere of New England, which he would return to throughout his career. He also studied the old masters from an early age, particularly Vermeer, whose mark is visible in the light and composition of many of his works. Barnet began his formal artistic studies at the School of the Boston Museum of Fine Arts, later moving to New York in 1931 to study at the Art Students League. He soon became the league’s official printer, and would continue printmaking throughout his career. Although his earliest work followed the trend of social realism, by the late 1940s, Cubist and other modernist experimentations gave way to an embrace of total abstraction, and throughout the 50s and 60s Barnet continued producing abstract work. In the 1960s he transitioned to the figural style for which he would be best recognized—his spacious, geometric scenes of family life (mostly his own), and portraits of pensive, solitary women in New England. Barnet would return to abstract work at the end of his life, starting in 2004 until his death in 2012. 

During his long career, Barnet taught at the Art Students League, Cooper Union and Yale University.  He was a member of the National Academy of Design and the American Academy of Arts and Letters, as well as a fellow of the Royal Society of Arts in London. In 2011, Barnet received the National Medal of Arts from President Obama in Washington DC.