  Let $1\le m<n$ be integers, and let $K\subset\mathbb{R}^{n}$ be a
self-similar set satisfying the strong separation condition, and with $\dim
K=s>m$. We study the a.s. values of the $s-m$-dimensional Hausdorff and packing
measures of $K\cap V$, where $V$ is a typical $n-m$-dimensional affine
subspace. For $0<\rho<\frac{1}{2}$ let $C_{\rho}\subset[0,1]$ be the attractor
of the IFS $\{f_{\rho,1},f_{\rho,2}\}$, where $f_{\rho,1}(t)=\rho\cdot t$ and
$f_{\rho,2}(t)=\rho\cdot t+1-\rho$ for each $t\in\mathbb{R}$. We show that for
certain numbers $0<a,b<\frac{1}{2}$, for instance $a=\frac{1}{4}$ and
$b=\frac{1}{3}$, if $K=C_{a}\times C_{b}$ then typically we have
$\mathcal{H}^{s-m}(K\cap V)=0$.
