Barker was a hard thrower, who early in his career struggled with his control. On April 16, 1978, at Fenway Park, Barker (then with the Texas Rangers) threw a pitch that sailed upward onto the screen above and behind the backstop. Partly due to this, he didn't make the majors for good until 1979.
Barker's best season statistically was 1980, when he enjoyed career-highs in wins (19) and strikeouts (181, best in the American League).
Barker's most notable accomplishment occurred on May 15, 1981 as a member of the Cleveland Indians. On a cold, damp night in Cleveland, Barker pitched the tenth official perfect game in baseball history, defeating the Toronto Blue Jays, 3–0 (the game was originally reported as the ninth perfect game in major league baseball history until the league later changed the criteria for recognizing a perfect game). The final out of the game was a fly ball caught by Rick Manning in short center field. Barker's pitching was so consistent that he never once reached ball three against any Blue Jay hitter.
Barker's perfect outing, one of only twenty-three in the history of Major League baseball, is also the last no-hitter thrown by an Indian. "I run into people almost every day who want to talk about it", Barker said in 2006. "Everyone says, 'You're probably tired of talking about it.' I say, 'No, it's something to be proud of.' It's a special thing."
Barker was selected for the 1981 Major League Baseball All-Star Game, held in Cleveland on August 9. The event was notable on two accounts: It was the first game played after a lengthy players' strike and gave Barker the opportunity to pitch two scoreless innings before 72,086 fans in his home stadium.
During the 1983 season, he was traded to the Atlanta Braves for Brett Butler, Brook Jacoby, Rick Behenna and $150,000 cash. The trade was initiated by the Braves, who were in a tight race for first in the National League West Division with the Los Angeles Dodgers. Barker pitched reasonably well down the stretch, notching a 3.82 ERA despite only going 1-3 in his six starts after the trade. After the season, the Braves signed Barker to one of the richest contracts for a pitcher in baseball history at the time--$4 million over five years.
However, the Braves didn't get nearly enough in return. In 1984, Barker went 7-8 with a 3.85 ERA before missing the last two months of the season with an elbow injury. The next year, Barker's ERA ballooned to 6.35, and he only managed a 2-9 record. He was released at the end of 1986 spring training with three years remaining on his contract. He signed with the Montreal Expos a few weeks later and spent the season with their top affiliate, the Indianapolis Indians. The Expos released him during 1987 spring training, and he finished his career with the Milwaukee Brewers. Meanwhile, Butler and Jacoby went on to become All-Stars.
After his playing career, Barker returned to the Cleveland area and founded a construction company with a business partner. He and his wife Eva are the parents of Jared, Blake and Jacob. The Barker family currently resides in Geauga County east of Cleveland. Barker serves as the head coach for Division II Notre Dame College in South Euclid.