Carr attended Phoenix Union High School where he competed in basketball, and representing P.U.H.S. as a discus thrower and in the shot put. Fred was in the top ten discus throwers in the USA in his senior year. He played for Phoenix College before transferring to the University of Texas at El Paso (UTEP) in 1965, where he was called "Probably the best overall linebacker in school history." Fred was inducted into the UTEP Athletics Hall of Fame, September 23, 2005, and would later also be elected to the Phoenix College (PCAA) Hall of Fame in their inaugural class of 2007.
Carr was drafted in the first round of the 1967 NFL Draft by the Green Bay Packers. Carr's professional career spanned 10 seasons, all with the Packers. He retired from pro football in 1977.