We prove that there are finitely many families, up to isomorphism in codimension one, of elliptic Calabi-Yau manifolds Y -> X with a rational section, provided that dim(Y) <= 5 and Y is not of product type. As a consequence, we obtain that there are finitely many possibilities for the Hodge diamond of such manifolds. The result follows from log birational boundedness of Kawamata log terminal pairs (X, Delta) with K-X + Delta numerically trivial and not of product type, in dimension at most four.