Let SI(Sg) denote the hyperelliptic Torelli group of a closed surface Sg of genus g. This is the subgroup of the mapping class group of Sg consisting of elements that act trivially on H1(Sg; ℤ) and that commute with some fixed hyperelliptic involution of Sg. We prove that the cohomological dimension of SI(Sg) is g − 1 when g ≥ 1. We also show that Hg−1(SI(Sg); ℤ) is infinitely generated when g ≥ 2. In particular, SI(S3) is not finitely presentable. Finally, we apply our main results to show that the kernel of the Burau representation of the braid group Bn at t = −1 has cohomological dimension equal to the integer part of n/2, and it has infinitely generated homology in this top dimension.