Calhoun went to Benicia High School in California. He was drafted in the 17th round of the 2013 Major League Baseball draft by the Tampa Bay Rays, however, he did not sign with them.
Calhoun played collegiate baseball for the Arizona Wildcats at the University of Arizona. He played collegiate summer baseball with the Wareham Gatemen of the Cape Cod Baseball League in 2014.
He moved to Yavapai College after a year in Arizona. Calhoun signed with Kansas State University out of junior college but chose to sign with the Dodgers.
Calhoun was traded to the Texas Rangers on July 31, 2017 (together with fellow minor leaguers A. J. Alexy and Bren Davis) in exchange for Yu Darvish.
On September 12, 2017, he was promoted to the big leagues for the first time. In 13 games, Calhoun hit.265/.324/.353/.677 with a home run and four RBI.
Calhoun batted.222/.269/.333/.602 with 2 home runs and 11 RBI in 35 big league games in 2018. He hit.294/.351/.431/.782 in 108 games with the Triple-A Round Rock Express, with 9 home runs and 47 RBI.
Calhoun suffered a broken forearm on June 27, 2021, after being struck by a pitch from Kansas City Royals pitcher Kris Bubic.