We construct isospectral pairs of Riemannian metrics on $S^5$ and on $B^6$, thus lowering by three the minimal dimension of spheres and balls on which such metrics have been constructed previously ($S^{n\ge8}$ and $B^{n\ge9}$). We also construct continuous families of isospectral Riemannian metrics on~$S^7$ and on~$B^8$. In each of these examples, the metrics can be chosen equal to the standard metric outside certain subsets of arbitrarily small volume.