Street played for South Kentucky College. He started his professional career with Hopkinsville in the Kitty League in 1890, playing for the pricely sum of $60 per month. Sold to Terre Haute in the Central League and then to Cincinnati, he made his big league debut in 1904.

He bounced around, going from Cincinnati to the Boston Braves and back to Cincinnati in 1905. The next two seasons he played with the San Francisco Seals of the Pacific Coast League. In 1908 he was sold to Washington and returned to the major leagues. With the Senators, Street became known as Walter Johnson's catcher, even though he played for Washington just four years, and the last two on a part-time basis. Street's best batting average during those four years was .222.

Johnson credited his success to Street for helping him maintain his concentration. Plus Street was not afraid to take the blame if one of his calls for a pitch ended up costing Washington the game.

"What a catcher he was," Johnson said, "a big fellow, a perfect target, a great arm, spry as a cat back of the plate, always talking, full of pep and fight. Gabby was always jabbering and never let a pitcher take his mind off the game. When we got in a tight spot Gabby was right there to talk it over with me. He never let me forget a batter's weakness." The young Johnson won 14 and 13 games in Street's first two seasons, then 25 in each of Street's last two seasons in Washington. By then, "the Big Train" was up to speed and Street was out of town.

During his stay with Washington, Street settled a bet between two sportsmen by catching a ball tossed from the top of the Washington Monument on the morning of August 21, 1908. (He missed 14 before he caught one. In 1894 catcher Pops Schriver had done it on the first try.) Although reported to be "considerably jarred by the impact," he pocketed a $500 prize and then caught Johnson's 3-1 victory over Detroit later that day.

Street was traded to the New York Highlanders in December 1911. From there he returned to the minors. When the United States entered World War I, Street enlisted, saw action in Europe, and left with the rank of sergeant. He was later called "Old Sarge."

Back in baseball as a manager in 1930, Street brought the Cardinals a pennant in his first year at the helm. They were defeated by the Philadelphia Athletics in six games in the World Series but repeated as pennant-winners in 1931. In a rematch with the A's, the Cards won in seven games. Nevertheless, St. Louis owner Sam Breadon fired Street in July 1933. The Cards had won only 11 of their last 33 games and were drawing less than 1,000 fans per contest. Street, seemingly unable to control the emerging Gas House Gang, particularly Dizzy Dean, was replaced by Frankie Frisch.