Binomial ideals are of interest in algebraic geometry because they correspond to toric varieties, which have many nice combinatorial properties. I will characterize the matrix Schubert varieties whose defining ideals are binomial in terms of pattern-avoidance. This proof uses Gröbner basis methods and generalizes a result of Escobar-Mészáros on matrix Schubert varieties that are toric with respect to the natural torus action.