The exact theorem states that if a single monkey types infinitely many consecutive symbols chosen randomly from a finite set containing all symbols in the complete works of Shakespeare, then with probability 1, he types the complete works. The proof goes as follows: let m be the size of the alphabet of symbols, and n be the length of the Bard's complete works. We can divide the monkeys typing into infinitely many blocks of length n. The probability that each block matches the complete works is (1/m)^n, which is a very small but nonzero number. If we make infinitely many attempts to do something with a small but positive chance of success, we will do it with probability 1, as the probability of not doing it over k trials goes to zero as k goes to infinity.
Alternatively, one could use the probabilities of the individual keys to construct a uniform probability measure on the set of all infinite strings; then the measure of the set of strings containing the complete works is 1. Of course, the monkey COULD type nothing but the letter a, but this scenario is not terribly likely.
The result is called the infinite monkey theorem, which has got to be the best title of any theorem in mathematics.
I'm not sure what exactly you mean by firing arrows at an infinite target, as one could imagine many ways of doing it which provide different probabilities of hitting the bullseye, but if the probability of hitting the bullseye in a single shot is zero, then it is a different situation entirely from the complete works, because the chance of typing the complete works in n letters, while exceedingly small, is not zero.