Hembree was drafted in the fifth round of the 2010 MLB draft by the San Francisco Giants. ;2010 Hembree began his professional career in 2010, pitching for the rookie leagueArizona League Giants and going 0–0 with a 0.82 ERA over 12 relief appearances, with 22 strikeouts and 3 saves in 11 innings pitched. ;2011 Hembree began the 2011 season pitching for the Class A-AdvancedSan Jose Giants, and was promoted to the Double-ARichmond Flying Squirrels in late June. He went a combined 1–1 with a 1.86 ERA over 54 relief appearances, with 78 strikeouts and 38 saves in innings. ;2012 During 2012, Hembree pitched for the San Jose Giants and the Triple-AFresno Grizzlies. Overall, he had a 4.19 ERA with 1–1 record, 15 saves, and 43 strikeouts in 43 innings pitched. ;2013 Hembree spent most of 2013 with Triple-A Fresno, where he appeared in 54 games, compiling a 4.07 ERA with 1–4 record, 31 saves, and 63 strikeouts is innings pitched. Late in the season, he was called up to the majors for the first time on September 3. He made his MLB debut that day, pitching a scoreless eighth inning in a 3–2 loss to the San Diego Padres. He appeared in a total of nine MLB games for the 2013 Giants during September, giving up no runs in innings pitched with 12 strikeouts. ;2014 Hembree started the 2014 season with Triple-A Fresno, where he registered 18 saves in 41 appearances, with 3.89 ERA, 1–3 record, and 46 strikeouts in innings pitched.
Hembree was traded to the Boston Red Sox, along with Edwin Escobar, for Jake Peavy on July 26, 2014. ;2014 Hembree made his Red Sox debut on August 9, wearing uniform number 38 and pitching four innings of scoreless relief in a 19-inning loss to the Los Angeles Angels; he gave up two hits, two walks, and struck out two. Due to the Red Sox signing Cuban outfielder Rusney Castillo on August 23, and guaranteeing Castillo the uniform number he wore in Cuba, which was 38, Hembree switched his uniform number to 37. For the remainder of the season, Hembree split time between Boston and the Triple-A Pawtucket Red Sox. With Pawtucket, he made seven appearances with 2.70 ERA with nine strikeouts in innings pitched, compiling an 0–1 record with two saves. He made a total of six MLB appearances for the 2014 Red Sox, striking out six in ten innings pitched, with a 4.50 ERA. ;2015 Hembree spent parts of 2015 with both Boston and Pawtucket. In Triple-A, he made 29 appearances, compiling an 0–5 record with 8 saves and 2.27 ERA, while striking out 32 in innings pitched. With the 2015 Red Sox, he made 22 MLB appearances, striking out 15 in innings pitched, while compiling a 3.55 ERA and 2–0 record. ;2016 Hembree again split time between Boston and Pawtucket during 2016. He made 13 Triple-A appearances, registering 8 saves with 0.68 ERA and 22 strikeouts in innings pitched. In 38 MLB appearances with the 2016 Red Sox, he compiled a 4–1 record with 2.65 ERA and 47 strikeouts in 51 innings pitched. ;2017 With the 2017 Red Sox for the entire season, Hembree made 62 appearances, striking out 70 in 62 innings pitched, with a 3.63 ERA and 2–3 record. ;2018 Hembree started the season as a member of the 2018 Red Sox, in his usual role as a member of Boston's bullpen. He finished 4–1 with a 4.10 ERA in 67 appearances, striking out 76 batters in 60 innings. Hembree was initially not included on Boston's postseason roster, but was added for the ALDS and beyond due to an injury to Steven Wright. Hembree made four appearances in the postseason, allowing no earned runs in innings, as Boston won the World Series over the Los Angeles Dodgers. ;2019 Hembree was included on Boston's Opening Day roster to start the 2019 season. He recorded his first MLB save on May 8, closing out an extra innings win over the Baltimore Orioles. Hembree was added to the injured list on June 14, retroactive to June 11, with a right elbow extensor strain; he returned to the team's active roster on July 4. He was again placed on the injured list on August 2, with right lateral elbow inflammation, and was activated on September 23. For the season with Boston, Hembree appeared in 45 games, recording a 3.86 ERA with 1–0 record and two saves, while striking out 46 in innings.
Personal life
Hembree and his wife Maci were married in 2017, they have two children.