Barrett graduated from Evansville Central High School and attended Wabash Valley College for two years before transferring to the University of Mississippi (Ole Miss).

[7] He made his debut in the ninth inning of the team's first game of the year against the New York Mets, striking out two while picking up his first career win.

[14] On July 23, 2016, while rehabbing at the Nationals' minor league complex in Viera, Florida, Barrett fractured the humerus bone in his right arm while pitching.

Teammate Mat Latos reportedly vomited in the dugout,[15] and Nationals manager Dusty Baker sequestered the only video of the injury on a locked hard drive so that no one could watch it.

[6] Barrett underwent surgery performed by Dr. James Andrews,[16] which left two plates and sixteen screws embedded in the bone of his arm.

He pitched against live competition for the first time since 2015 during a spring exhibition against the Canadian national baseball team[17] and was assigned to the Short Season-A Auburn Doubledays to start the 2018 season.

[21] After spending part of spring training on the major league side of the Nationals camp in West Palm Beach, Florida,[15] Barrett was assigned to Class-AA Harrisburg to start the 2019 season.

[22] As the season progressed, Barrett built up arm strength and recovered much of his former velocity, also incorporating a changeup into a pitching arsenal that had predominantly featured a fastball and a slider.

[25] Barrett appeared in a game for the first time since the 2015 season on September 7, pitching a scoreless inning in relief against the Atlanta Braves and picking up his first strikeout since August 1, 2015, as he caught All-Star Ronald Acuña Jr. looking on a slider.

The Nationals finished the year with a 93–69 record, clinching a wild card spot and eventually winning the 2019 World Series over the Astros.

Barrett spent the 2021 season in the minors, splitting time between the Triple-A Rochester Red Wings, Double-A Harrisburg Senators, and the Single-A Fredericksburg Nationals, posting a 2.13 ERA in 33 total appearances.

[30] Shortly after his retirement, Barrett announced that he would be staying with the Triple-A Lehigh Valley IronPigs to serve as the team's bullpen coach for the remainder of the 2022 season.