Bairstow was born and raised in Brisbane, Australia. He attended Runcorn State High School before graduating from Anglican Church Grammar School in 2008. His parents are Ian and Penny Bairstow, a veterinarian and a school teacher, who both also coach youth basketball in Australia. Bairstow is the third of seven children, with four brothers (Kieran, Jarred, Daniel, and Sean) and two sisters (Melissa and Stephanie). Three of his siblings have also played college basketball: brother Jarred at the University of Central Oklahoma, sister Stephanie at Utah State University, and brother Sean at Utah State University.

Bairstow played for the South West Metro Pirates and Brisbane Capitals youth teams before attending the Australian Institute of Sport (AIS) in Canberra in 2009 and 2010. He played two seasons with the AIS in the South East Australian Basketball League (SEABL), and in 2010, he was a member of the SEABL Select Team that visited Qatar. He was also on the Queensland team that won the 2009 Under 20 Australian national championship, and he played on the Australian Under 19 team that won the 2009 FIBA Oceania title in Saipan.

As a sophomore, Bairstow became a regular rotation player, appearing in 34 games, averaging 3.7 points and 3.6 rebounds in 15.4 minutes per game. Fellow Australian Hugh Greenwood joined the Lobos as a freshman in 2011–12 – he and Bairstow were teammates at the AIS in 2009 and 2010. Gordon had a dominating senior season, while Williams and Snell each averaged in double figures. The Lobos finished the season 28–7 after earning a share of the regular season Mountain West Conference (MWC) title, winning the conference tournament championship, and appearing in the NCAA Tournament.

Considered a two-star recruit by ESPN.com, Bairstow was listed as the No. 196 power forward in the nation in 2010.

Following the 2010 SEABL season, Bairstow moved to the United States to attend the University of New Mexico. In his freshman season, the young Lobo squad was led by senior Dairese Gary and transfer Drew Gordon and finished the year 22–13 with an appearance in the NIT. Bairstow was a member of one of the most accomplished freshman recruiting classes in Lobo history, along with Kendall Williams, Tony Snell and Alex Kirk. He appeared in 31 games but played sparingly, averaging 2.6 points and 1.8 rebounds in 9.7 minutes per game.

In July 2012, Bairstow was named to the Australian national team, the Boomers, for the 2012 Australian Stanković Cup, and the team went on to a second-place finish. After his 2012–13 college season, he was named to the Boomers for the Sino-Australia Challenge against China in June 2013. He excelled during his debut with the Boomers and carried that form into the 2013 Stanković Cup and World University Games, winning gold and silver respectively. In August 2013, he was named to the Boomers 2013 FIBA Oceania Championship squad to take on New Zealand in a two-game series.

Bairstow gained attention during his senior season for his ritual of going to the weight room to lift after games, while still in uniform, receiving praise from opposing coaches and the media for his commitment and work ethic. When he arrived at New Mexico, he measured 6–8 and 210 pounds; by his senior season he was listed as 6–9 and 250 pounds; at the NBA Draft Combine in May 2014, he surprised observers by measuring just under 6–10, suggesting he had still been growing. Bairstow was arguably the most improved player in the history of the Lobo program. Head coach Craig Neal stated, “I don’t think there’s anybody who’s come farther in four years. I don’t think there’s anybody who could have predicted what he’s done this season.”

Bairstow was selected by the Chicago Bulls with the 49th overall pick in the 2014 NBA draft, and subsequently signed a three-year contract with the Bulls, with one and a half years guaranteed. He made his NBA debut on 29 October 2014 in the Bulls’ season opener against the New York Knicks, recording one turnover and missing his only field goal attempt in three and half minutes. He appeared in 18 of the Bulls’ 82 regular-season games, making one start. In his one start on 25 November against the Denver Nuggets, he played a season-high 18 minutes and 46 seconds and had two points, two rebounds, one assist and one block, but also accumulated four fouls and two turnovers and made just one of four field goal attempts. He failed to appear in any of the Bulls’ 12 playoff games.

In July 2014, Bairstow was named to the Boomers squad for the 2014 FIBA Basketball World Cup in Spain.

The 2015–16 season saw Bairstow once again appear in just 18 regular-season games, while making two starts. He played in five games on assignment with the Austin Spurs in the NBA Development League between 22 November and 7 December. On 2 March 2016, he scored a career-high eight points in 10 minutes as a starter in a 102–89 loss to the Orlando Magic.

On 17 June 2016, Bairstow was traded to the Detroit Pistons in exchange for Spencer Dinwiddie. He was waived by the Pistons on 10 July 2016.

On 18 July 2016, Bairstow signed with his hometown team, the Brisbane Bullets of the National Basketball League. Due to a shoulder injury he sustained while playing for the Boomers at the Rio Olympics, Bairstow missed the Bullets’ entire preseason schedule, as well as the first two games of the regular season. In his NBL debut on 16 October, Bairstow recorded 15 points, seven rebounds and three assists off the bench in a 96–93 overtime loss to the Adelaide 36ers. On 20 December, he was ruled out for the rest of the season after scans revealed an anterior cruciate ligament injury to his left knee. He suffered the injury during overtime in Brisbane’s 91–85 loss to the Illawarra Hawks the previous night. Having never previously suffered a serious injury, Bairstow had now copped two in the space of six months, including a career-threatening knee injury. In 13 games over the first two months of the 2016–17 season, he averaged 12.8 points and 6.5 rebounds per game.

In July 2016, Bairstow was named to the Boomers squad for the 2016 Summer Olympics in Rio de Janeiro, Brazil. There, he dislocated his shoulder in the pool game against Venezuela.

After missing the entire 2017–18 season with the injury, Bairstow re-signed with the Bullets on a two-year deal on 4 May 2018. He returned to action in the 2018–19 season. In March 2019, the Bullets elected not to take up the option on Bairstow’s contract.

On 25 July 2019, Bairstow signed with Lithuanian team Rytas Vilnius.

On 4 August 2020, Bairstow signed with The Hawks for the 2020–21 NBL season, returning to the league for a second stint.