I’d like to see an All College team (all with at least four-year degrees) and an All Unschooled team (preferably nobody who even finished high school; bonus if they are of good character) 
And an all-military team–players who served in the armed forces with distinction.
Many great players left off this list, but here’s a shot at an All-Military team. Jim Kern is the only name on the list that debuted in the majors after 1960:
C - Yogi Berra
1B - Hank Greenberg
2B - Eddie Collins
SS - Ernie Banks
3B - Jackie Robinson
OF - Ted Williams
OF - Willie Mays
OF - Stan Musial
Bench - George Sisler, Joe DiMaggio, Tris Speaker, Luke Appling, Mickey Cochrane
SP - Bob Feller
SP - Warren Spahn
SP - Grover Cleveland Alexander
SP - Christy Mathewson
SP - Ted Lyons
RP - Hoyt Wilhelm
RP - Red Ruffing
RP - Jim Kern
Manager - Red Schoendienst
Ralph “The Major” Houk would make a good manager, too.
As would Tug McGraw, who even threw a pretty fair country screwball.