  We show that non-trivial $\times N$-invariant sets in $[0,1]^d$, such as the
Sierpi\'{n}ski carpet and the Sierpi\'{n}ski sponge, are tube-null, that is,
they can be covered by a union of tubular neighbourhoods of lines of
arbitrarily small total volume. This introduces a new class of tube-null sets
of dimension strictly between $d-1$ and $d$. We utilize ergodic-theoretic
methods to decompose the set into finitely many parts, each of which projects
onto a set of Hausdorff dimension less than $1$ in some direction. We also
discuss coverings by tubes for other self-similar sets, and present various
applications.
