Mike Greenwell played his entire 12-year career (1985-96) with the Red Sox. 

Larry Gardner debuted for the Red Sox in 1908, and he was affixed as the starting Second Baseman in 1910.  Gardner moved to Third the year after, and he was their starter until he was traded to the Philadelphia Athletics after the 1917 season.  He was a steady presence, helping them win three World Series Championships (1912, 1915 & 1916), and he would have two .300 seasons. 

Pete Runnels was a good infielder during his time with the Washington Senators, but when he joined the Red Sox, he took a lot more than his versatile glove with him, as he found far more plate discipline.

How good an athlete was Jackie Jensen?  He was the first person to play in the Rose Bowl, a World Series, and an MLB All-Star Game.