Gobert was born in Saint-Quentin, Aisne, in the north of France. He is the son of Corinne Gobert, and Rudy Bourgarel, a former professional basketball player from Guadeloupe who played for the French national team in the 1980s. Prior to his professional career, Bourgarel had attended Marist College in New York. Raised in his native town, Gobert started playing basketball in 2003 for the JSC St-Quentin club, before later joining the Saint-Quentin Basket-Ball club. In 2007, he joined the Cholet training center and in 2010, he participated in the FIBA Europe Under-18 Championship for the French junior national team where he finished as the team's best scorer and rebounder.
From 2009 to 2011, Gobert mostly played for the Cholet Basket junior team, managing just one game for the senior team in 2010–11, recording 6 points, 5 rebounds, 1 steal and 1 block. In 2011–12, he averaged 4.2 points and 3.6 rebounds per game in 29 games for Cholet. During the next season in 2012–13, he averaged 8.4 points and 5.4 rebounds per game in 27 games.
Gobert declared for the 2013 NBA draft and set NBA Draft Combine records for wingspan (7 feet 8½ inches) and standing reach (9 feet 7 inches) in the 2013 combine. These dimensions earned him the nickname "The Stifle Tower." The record for wingspan was broken a year later by Walter Tavares.
On June 27, 2013, Gobert was selected with the 27th overall pick by the Denver Nuggets. He was later traded on draft night to the Utah Jazz. On July 6, he signed his rookie scale contract with the Jazz and joined them for the 2013 NBA Summer League. In just his 12th game for the Jazz on November 24, Gobert scored a season-high 10 points against the Oklahoma City Thunder. On December 14, he was assigned to the Bakersfield Jam of the NBA Development League. He was later recalled by the Jazz on December 21, reassigned on January 4, and recalled again on January 13. He played sparingly for the Jazz in 2013–14, making an appearance in 45 of the team's 82 regular season games.
In July 2014, Gobert re-joined the Jazz for the 2014 NBA Summer League. On October 24, 2014, the Jazz exercised their third-year team option on Gobert's rookie scale contract, extending the contract through the 2015–16 season. On January 9, 2015, he recorded a career-high seven blocked shots in a loss to the Oklahoma City Thunder. On March 3, he recorded 15 points and a career-high 24 rebounds in a 93–82 win over the Memphis Grizzlies. His 24 rebounds were three short of the Jazz franchise record of 27 set twice by Truck Robinson, both in the 1977–78 season. In April 2015, he twice scored a career-high 20 points, and finished the 2014–15 season with 25 double-doubles. He went on to finish third in the NBA Most Improved Player Award voting.
On October 17, 2015, the Jazz exercised their fourth-year team option on Gobert's rookie scale contract, extending the contract through the 2016–17 season. On December 2, 2015, he was ruled out indefinitely after sustaining a sprained left MCL. He returned to action on January 7, 2016 against the Houston Rockets after missing 18 games with the injury. He was limited to just 15 minutes but chipped in six points and three rebounds. On January 16, he had a season-best game with 18 points, 18 rebounds and 5 blocks in a 109–82 win over the Los Angeles Lakers. On February 27, he recorded 12 points, a season-high 19 rebounds and six blocks in a 98–96 loss to the Brooklyn Nets.
On October 31, 2016, Gobert signed a four-year, $102 million contract extension with the Jazz, which at the time made him the best paid French athlete ever (in terms of salary per year). On December 6, 2016, he scored a career-high 22 points in a 112–105 win over the Phoenix Suns. Four days later, he had 17 points, 14 rebounds and a season-high six blocks in a 104–84 win over the Sacramento Kings. On January 10, 2017, in a win over the Cleveland Cavaliers, Gobert posted his franchise-best 25th consecutive game with 10-plus rebounds (14). On January 20, 2017, he recorded career highs of 27 points and 25 rebounds in a 112–107 overtime win over the Dallas Mavericks. The following day, he had his 31st double-double of the season with 19 points and 11 rebounds in a 109–100 win over the Indiana Pacers. His 11 rebounds marked his 30th straight game with at least 10 rebounds—only four other NBA players have had a 30-game streak of 10-rebound games since 1997–98. His streak ended at 30 after falling one rebound shy of his 31st consecutive 10-plus rebound game on January 23 against the Oklahoma City Thunder. With two blocks against New Orleans on February 8, Gobert reached 499 career blocked shots, passing Rich Kelley (498) for ninth on the franchise's all-time block list. On March 5, he had 16 points and 24 rebounds and tipped in a missed shot as time expired in overtime to lift the Jazz to a 110–109 win over the Sacramento Kings. On March 20, he had 16 points, 14 rebounds and a career-high eight blocks in a 107–100 loss to the Indiana Pacers. Two days later, he scored a career-high 35 points and grabbed 13 rebounds in a 108–101 win over the New York Knicks. Eleven of his 13 rebounds were offensive.
In Game 1 of the Jazz's first-round playoff series against the Los Angeles Clippers on April 15, Gobert suffered a hyperextension and bone bruise to his left knee; the injury occurred on the first possession of the game. Utah took Game 1 without him, but he went on to miss Games 2 and 3, as the Jazz went down 2–1 in the series. Gobert returned to the line-up for Game 4 and contributed to a 105–98 win with 15 points and 13 rebounds. The Jazz went on to eliminate the Clippers with a 104–91 victory in Game 7, closing out the first-round series 4–3 to earn the franchise's first postseason victory since 2010. However, they were swept by the Golden State Warriors in the second round to bow out of the playoffs. At the season's end, Gobert was named in the All-NBA Second Team for the first time in his career. He also earned NBA All-Defensive First Team honors.
Gobert won the bronze medal at the 2011 FIBA Europe Under-20 Championship and the silver medal at the 2012 FIBA Europe Under-20 Championship. In the 2012 edition, Gobert was named to the All-Tournament team alongside countryman, Léo Westermann.
Gobert was called by coach Vincent Collet to help the French national team prepare for the 2012 Summer Olympics tournament. He played two friendly games, scoring 8 points. He was part of the team that won the bronze medal at the 2014 FIBA Basketball World Cup, where he averaged 4.1 points per game.
In 2015, he helped France to bronze at the European Championships, averaging 10.4 points, 8.1 rebounds and 2.0 blocked shots during the tournament, while receiving Eurobasket.com All-European Championships Second Team distinction.
At the 2016 Olympic Games in Rio, Gobert played in five games for France, tallying 5.7 points as well as 7.2 boards and 1.8 blocks per outing.