Left-handed hitters have two large advantages in baseball. A left-handed batter is closer to first base when he hits the ball, and therefore is able to get there faster. A breaking ball from a right-handed pitcher breaks in towards a left-handed hitter, making it easier to hit.
However, left-handed throwers are, by the nature of the game, restricted to pitcher, first base, and the outfield. For a left-handed second baseman, the double play pivot is near-impossible, as is making a throw on any ball hit up the middle or dribbled in front. For a left-handed shortstop, the throw from the third base hole is impossible, as is the throw on a dribbler. For a left-handed third baseman, the throw on any ball hit over the bag is impossible, as is the throw on a bunt.
The case against left-handed catchers is less severe, though the throw trying to catch a runner stealing 3rd would be difficult to make in time. (Watch any catcher trying to pick a runner off at 1st; he needs to completely turn his body to do so.) Also, he would more frequently be throwing into the batter.
Very rarely, a lefty has been pressed into service at one of those four positions. The next four posts list and detail each time that has occurred in the major leagues in the past 85 years.
I restricted the lists to games from 1920 and later. The use of lefties at odd positions was much more common (and therefore less interesting) in 19th century and early 20th century baseball, and searching for pre-1957 box scores is really time-consuming, especially when one doesn’t know which game to look for.