The proportion ρk of gaps with length k between square-free numbers is shown to satisfy logρk=−(1+o(1))(6/π2)klogk as k→∞. Such asymptotics are consistent with Erdős's challenge to prove that the gap following the square-free number t is smaller than clogt/log logt, for all t and some constant c satisfying c>π2/12. The results of this paper are achieved by studying the probabilities of large deviations in a certain ‘random sieve’, for which the proportions ρk have representations as probabilities. The asymptotic form of ρk may be obtained in situations of greater generality, when the squared primes are replaced by an arbitrary sequence (sr) of relatively prime integers satisfying [sum ]r1/sr<∞, subject to two further conditions of regularity on this sequence.