Born in Tishomingo, Oklahoma, Murrah ran away from home at the age of 13, riding freight trains until he was dropped in the community of Tuttle, Oklahoma. A local family took him in and housed him, in exchange for labor on their farm. Murrah worked his way through high school and college, ultimately earning a law degree from the University of Oklahoma.
In 1937, at the relatively young age of 32, Murrah was appointed to the U.S. District Court for Oklahoma, making him one of the youngest federal judges in American history. In 1940, he rose to the 10th Circuit Court of Appeals, where he served until his retirement in 1970. From 1969 to 1974, he directed a federal judicial center in Oklahoma City.
U.S. Supreme Court Chief Justice Warren Burger referred to Murrah as "one of the foremost figures in the American judiciary." He is noted for having pioneered the practices of pretrial conferences and of litigation panels with judges from multiple judicial districts.
Murrah identified politically with the Democratic Party. A Methodist, Murrah taught Sunday School classes at Oklahoma City's Crown Heights Methodist Church, of which he was a major financial supporter. Murrah was also a Freemason.