Born in a sawmill camp known as Yellow Pine, near Many, Sabine Parish, Louisiana, Gossett moved to Texas in 1908 with his parents, who settled on a farm near Henrietta, Clay County. He attended the rural schools of Clay and Garza Counties, Texas. He was in the University of Texas at Austin, A.B., 1924 and the law school of the same university, LL.B., 1927. He was admitted to the bar the latter year and commenced practice in Vernon, Texas. He moved to Wichita Falls, Texas, in 1937 and continued the practice of law. He served as district attorney of the forty-sixth judicial district 1933-1937.
Gossett was elected as a Democrat to the Seventy-sixth and to the six succeeding Congresses and served from January 3, 1939, until his resignation July 31, 1951. He served as chairman of the Committee on Elections No. 2 (Seventy-seventh through Seventy-ninth Congresses). He resumed the practice of law and was general attorney for the Texas Southwestern Bell Telephone Co.. He served as judge of Criminal District Court, Dallas, Texas, until his death on November 6, 1990.