Eddie "The Fiddler" Basinski had 20/800 vision. He wore glasses so thick, his coach wouldn't let him play on the high school baseball team. He studied engineering at the University of Buffalo, which had no baseball team, but he played tennis and ran for the cross country team. He was a near virtuoso on the piano and a concert violinist. During World War II, Dodgers GM Branch Rickey was looking for promising players who were classified 4-F in the military draft. Basinski had been playing baseball in city leagues and was sent to work out with the Dodgers. They were short-handed that day, they played him in a major league game, and on his first swing, he hit a triple off the center-field wall. When the real major leaguers came back from the war, Basinski could not compete, but he played 12 more seasons professionally, and is in the Pacific Coast League Hall of Fame.
There are many stories of players overcoming obstacles on their way to the major leagues. Outfielder Dummy Hoy was deaf, but he played in the big leagues from 1888-1902, accumulating 2,044 hits. Pitcher Monty Stratton lost his right leg in 1938 in a hunting accident, ending his major league career, but played in the minor leagues from 1946-53 with a prosthetic leg. Outfielder Pete Gray batted .218 for the St. Louis Browns in 1945 despite having lost his right arm in a truck accident when he was 6. Three Finger Brown pitched from 1903-16, won 239 games with a 2.06 ERA and made it to the Hall despite losing two fingers on his right (pitching) hand in a farm-machinery accident as a youth.
Walter Johnson, the greatest pitcher ever, played in only two games in high school, one as a catcher, one as a pitcher, and he was terrible in that one. But he joined a town team, and after joining a second town team, this one in Idaho, he went on to win 417 games in the big leagues.
Bob Feller's father painted a bull's-eye on the side of the family barn in Van Meter, Iowa. Young Bob wore out that side of the barn with an upper-90s fastball and was signed by the Indians at age 17. He left high school to pitch for the Indians -- in one game, he struck out more than his age -- then after the season, he returned to finish his senior year. His high school graduation was broadcast on national radio.
More recently, there are dozens of stories of players who came from nowhere to star in the major leagues. Mike Piazza was a 62nd-round draft pick of the Dodgers -- Los Angeles manager Tommy Lasorda selected him as a favor to the family -- and went on to become the greatest hitting catcher of all time. Reggie Sanders was more of a track star in college, then concentrated on baseball, and went on to hit 300 homers and steal 300 bases in the major leagues. Speedy Jeff Stone was found by the Phillies after running barefoot through the corn fields in his home state of Missouri. Kenny Rogers was an outfielder in high school but was signed as a pitcher because of his strong throwing arm. The Rangers brought him to spring training as a pitcher. Pitching coach Sid Hudson asked Rogers to go from the stretch, and Rogers said, "I don't know how to do that." Rogers went on to win 219 games in the major leagues; included was a perfect game.