Several sources I've encountered on the World Wide Web say that the phrase "back to square one" meaning "back to the beginning" originates in early BBC radio commentaries for soccer/football -- to make things clearer for the listener, the field was divided into numbered squares, and the goalkeeper stood in square #1. (I'm not sure how much I trust these sources.)

Tem42 says the phrase "more likely started in the early 1900s, from the game of hopscotch, where if you make a mistake, you go back to the first square and start again." This makes a lot more sense to me, but I cannot find any support for it in my research; the trivia game at www.boardgame.co.uk/porky_pies-full.pdf is the only one that even mentions this possibility, but states that the soccer one is the correct origin.