Born in Roxbury, Massachusetts, Varney attended North Quincy High School and Deerfield Academy before enrolling at Harvard, where he played varsity football as well as baseball.

In the 85th Harvard–Yale game on November 23, 1968, tight end Varney caught Frank Champi's pass for the two-point conversion in the final second to earn a tie, and a share of the Ivy League championship, with Yale.

[1] After being drafted six previous times by five different Major League Baseball teams, Varney signed with the Chicago White Sox after they selected him in the first round of the secondary phase of the 1971 Major League Baseball Draft following his graduation from Harvard.

He spent much of the rest of that season with the Triple-A Richmond Braves, coming to bat for Atlanta ten times, with one hit, a single.

All told, in 69 MLB games played, Varney batted .247, with seven doubles and one triple, along with his five homers.