Barron was born in Mobile, Alabama. He attended St. Paul's Episcopal School in Mobile, where he played for the St. Paul's Saints high school football team, under the coaching of Eddie Guth. He played multiple positions, including running back, wide receiver, and linebacker from 2004 through 2007. Barron had 785 yards rushing as a junior on 78 carries and added 368 receiving yards on 23 catches, scoring a total of 12 touchdowns. As a senior, Barron was the MVP of the 2007 Class 5A state championship game after rushing for 1,094 yards on 122 carries with 15 touchdowns in his senior season. He was also named first-team Alabama Sports Writers Association All-State selection in Class 5A as a senior.
In addition to playing football, Barron also competed in track & field, where he won multiple 5A individual state titles in events ranging from the shot put to the long jump during the 2008 season. He recorded a career-best throw of 17.09 meters in the shot put at the Mobile Challenge of Champions, where he placed first. He captured four state titles at the 2008 AHSAA 4A-6A Championships, winning the triple jump, with a jump of 14.08 meters, long jump, with a leap of 6.74 meters, discus throw, with a throw of 44.03 meters, and in shot put, with a throw of 16.76 meters. He also recorded a personal-best time of 11.39 seconds in the 100 meters
Barron was ranked as a five-star prospect and the number one weakside linebacker in the 2008 class by scout.com. He was also ranked as the 5th best player in the state of Alabama and the 55th best player in the nation in the class of 2008 by rivals.com. Barron chose to play at Alabama over offers from Auburn, LSU, Florida, Florida State and Tennessee among others.
As a true freshman for the 2008 Crimson Tide, he made his first appearance on kick-off coverage against Clemson. He had his first career tackle early in the game on a crushing blow of Jacoby Ford on a kickoff return. Barron played in all of Alabama's 14 games, recording 18 tackles and a sack.
During his sophomore season in 2009, Barron took over as a starter at safety and was second on Alabama's national championship team with 74 tackles and an Southeastern Conference-leading seven interceptions and a touchdown. He also had a career high two interceptions against Mississippi State. Barron finished the regular season tied for third in the nation with 7 interceptions and tied for sixth in the nation with 17 passes defended. He was recognized as a first-team All-SEC selection and third-team All-American at safety.
Barron was the Crimson Tide's only returning starter in the defensive secondary for the 2010 season. He finished the season with 77 tackles and three interceptions, and was named an FWAA All-American. In 2011, he was again a first-team All-SEC selection and was recognized as a unanimous first-team All-American.
Barron was drafted with the 7th overall pick by the Tampa Bay Buccaneers in the 2012 NFL Draft. He is the highest selected Alabama defensive back in school history, surpassing Antonio Langham, who was chosen ninth overall in 1994.
"Boy, is this a solid pick. I love this kid because his floor and ceiling are the same thing. His ceiling is to be a Pro Bowl safety; his floor is to be a Pro Bowl safety. That's how good this young man is. This is going to be a good football team next year."
-Mike Mayock, former NFL safety and current NFL Network analyst
On July 20, 2012, Barron agreed to a 5-year deal with the team worth $14.5 million over four years, with no offset language. Barron was expected to be the strong safety starter his rookie season. In Week 3 of the preseason, Barron intercepted Tom Brady for his first career NFL touchdown. Barron started the regular season strong with 42 tackles, seven passes defended and one interception in his first six NFL games, but trailed off towards the end of the season before finishing with 89 tackles. He started all 16 games and made the Pro Football Writers Association All-Rookie team, even though the team was the NFL's worst pass defense.
On October 28, 2014, the Buccaneers traded Barron to the St. Louis Rams for a fourth and sixth-round draft pick in the 2015 NFL draft. In 2015, Barron switched over to the weakside linebacker position, after an injury to Alec Ogletree.