Barnes was born in Riverside, California and attended Riverside Polytechnic High School. He played college baseball at Arizona State University for the Arizona State Sun Devils from 2009 to 2011. Before his sophomore season in 2010 he moved from infield to catcher. During his career he played in 109 games and hit .308/.379/.429 with two home runs.

On December 10, 2014, Barnes was traded to the Los Angeles Dodgers, along with Chris Hatcher, Andrew Heaney, and Enrique Hernández, in exchange for Dan Haren, Dee Gordon, Miguel Rojas and cash. He was assigned to the AAA Oklahoma City Dodgers. On May 23, 2015, he was called up to the Major Leagues for the first time when Yasmani Grandal went on the 7-day DL with a concussion.

Barnes made his MLB debut as the starting catcher for the Dodgers on May 24, 2015, originally wearing number 65. He had one hit in three at-bats in his debut, with his first MLB hit being a single to center field off of Dale Thayer of the San Diego Padres. He returned to Oklahoma City after his short time on the Dodgers roster and was named as a starter on the Pacific Coast League team for the mid-season Triple-A All-Star. He was also named to the post-season PCL all-star team and Baseball America’s Triple-A All-Star team. He rejoined the Dodgers in September. He played in 20 games for the Dodgers with six hits in 29 at-bats (.207). In 81 games for the Oklahoma City team, he hit .315 with nine homers and 42 RBI.

Barnes appeared in 21 games for the Dodgers in 2016, hitting .156, and 85 games for Oklahoma City, hitting .295. He changed his number from 65 to 28. He also appeared in two games in the 2016 National League Division Series against the Washington Nationals, with one pinch hit appearance and one pinch run appearance.

Barnes made the Opening Day roster as the backup catcher in 2017. On June 30, 2017, he recorded his first multi-homer game of his career (his first career grand slam and a three run home run) and recorded a career-high seven RBIs against the San Diego Padres. He gradually saw more playing time, moving into a platoon role with Grandal for most of the season before taking over as the starter in the playoffs. During the regular season, he had a .289 batting average with eight home runs and 38 RBI in 102 games. In the 2017 NLDS Barnes had four hits in eight at-bats with a home run and a double in the Dodgers’ three-game sweep of the Arizona Diamondbacks. In the 2017 NLCS against the Chicago Cubs, he had two hits in 15 at-bats, and in the 2017 World Series against the Houston Astros, he had four hits in 23 at-bats.

In 2018, Barnes started spring training battling elbow discomfort. He ended up being the backup to Yasmani Grandal, taking over the catching role more in the postseason when Grandal allowed two passed balls and made two errors in game 1 of the NLCS and another passed ball in game 3, resulting in Dodgers fans booing him and begging for Barnes to catch. Barnes hit .205/.329/.290 in 100 games with 4 home runs, 41 hits, and 14 RBI. An elite framer, he excelled defensively. He ranked third in the MLB for framing and defensive statistics, had a .993 fielding percentage, and allowed only one passed ball. In the 2018 NLCS against the Milwaukee Brewers, Barnes had two hits in 18 at-bats. In the 2018 World Series against the Boston Red Sox, he was hitless in eleven at-bats.

In 2019, Barnes was named the Opening Day starting catcher after Grandal signed with the Milwaukee Brewers. However, he got off to a slow start and on July 26 he was sent down to Oklahoma City to make room for rookie catcher Will Smith to take over the everyday catching duties. Barnes was batting .196 prior to being sent down. He appeared in only 75 games in the majors in 2019, hitting .203/.293/.340 with five home runs and 25 RBIs, while he hit .264 in 23 games in the minor leagues. Barnes signed a one-year, $1.1 million, contract with the Dodgers after the season, avoiding arbitration.

During the pandemic-shortened 2020 season, Barnes appeared in 29 games for the Dodgers, hitting .244/.353/.314 with one homer and nine RBIs. In the postseason, Barnes was the starting catcher for each of Clayton Kershaw’s starts and in the World Series caught Walker Buehler as well. He had two hits in three at-bats in the Wild Card Series, two hits in two at-bats in the NLDS and two hits in seven at-bats in the NLCS. In the 2020 World Series against the Tampa Bay Rays, he had two hits in 13 at-bats. In Game 3, he drove in a run with a safety squeeze and also hit a home run, becoming only the second player to do both of those things in the same World Series game, joining Al López in 1961. The Dodgers won the World Series in six games.