Stassi attended Yuba City High School in Yuba City, California. He played for his school's baseball team, and was twice named The Sacramento Bee's player of the year. Stassi also competed for the United States national baseball team, winning gold in the 2006 Pan American Games in Barquisimeto, Venezuela, and the International Baseball Federation's 16 and Under Baseball World Championship in 2007. He finished his high school career with a .514 batting average, 40 home runs, and 162 runs batted in. Stassi committed to attend the University of California, Los Angeles (UCLA), to play college baseball for the UCLA Bruins.
Heading into the 2009 Major League Baseball (MLB) Draft, Sporting News called Stassi the best available high school catcher. Stassi made it clear that he expected a large signing bonus to agree to a professional contract. A potential first round pick, Stassi was not selected in the first three rounds, which Baseball America called "one of the biggest surprises" of the draft. The Oakland Athletics selected Stassi in the fourth round, with the 123rd overall selection, of the draft. He received a $1.5 million bonus, a record for a fourth round pick, forgoing his commitment to UCLA. Stassi also negotiated in his contract that the Athletics would invite him to spring training with the major league team in 2010.
Stassi played for the Vancouver Canadians of the Class A-Short Season Northwest League, and had a .280 batting average. In 2010, he performed well at spring training, impressing team officials, who assigned him to the Kane County Cougars of the Class A Midwest League. He played for the Stockton Ports of the Class A-Advanced California League in 2011, but missed the end of the season with shoulder surgery. The Athletics invited him to spring training in 2012. He batted .268 with 15 home runs and 45 RBIs in 84 games with Stockton during the 2012 season.
After the 2012 season, the Athletics traded Stassi to the Houston Astros with Brad Peacock and Chris Carter for Jed Lowrie and Fernando Rodriguez. Playing for the Corpus Christi Hooks of the Class AA Texas League in 2013, Stassi hit home runs in five consecutive games in July. He was named the Texas League's player of the week twice in July, and was named the Texas League's player of month for July.
The Astros promoted Stassi to the majors on August 20, 2013, to serve as a backup to Jason Castro when the Astros placed Carlos Corporán on the disabled list. Making his MLB debut that day, he recorded two hits. In his second MLB game, Stassi was hit in the face by a 96 miles per hour (154 km/h) fastball, which resulted in his being placed on the disabled list. Since the bases were loaded at the time he was plunked, he earned his first major-league RBI. Stassi began the 2014 season with the Oklahoma City RedHawks of the Class AAA Pacific Coast League (PCL). He finished the season with the Houston Astros, batting .350 (7 for 20) and driving in four runs in seven games played.
Stassi played for the Fresno Grizzlies of the PCL in 2015, before being promoted to the major leagues in August due to an injury to Castro. He played in 11 games for the Astros. Following the offseason trade of Hank Conger, Stassi entered spring training in 2016 expected to become the backup to Castro. In March, he had surgery on a fractured hamate bone in his left wrist. The Astros activated Stassi from the disabled list in May and optioned him to Fresno.
The Astros removed Stassi from their 40-man roster in March 2017. He played for Fresno before the Astros promoted him to the major leagues when Brian McCann was injured on August 14. Stassi went on the disabled list on August 25 with left hand inflammation caused by a blister.
Stassi's father, grandfather, and great-grandfather played as catchers in minor league baseball. His father, Jim, coached him at Yuba City. Max's great uncle, Myril Hoag, played in three World Series, batting .320 in them. His brother, Brock Stassi, is a major league player for the Philadelphia Phillies. Max and Brock Stassi have raised more than $100,000 with an annual charity event held in Yuba City, called "Homers for the Hungry." The money goes to local food banks.