  The Maxwell-Dirac system describes the interaction of an electron with its
self-induced electromagnetic field. In space dimension $d=3$ the system is
charge-critical, that is, $L^2$-critical for the spinor with respect to
scaling, and local well-posedness is known almost down to the critical
regularity. In the charge-subcritical dimensions $d=1,2$, global well-posedness
is known in the charge class. Here we prove that these results are sharp (or
almost sharp, if $d=3$), by demonstrating ill-posedness below the charge
regularity. In fact, for $d \le 3$ we exhibit a spinor datum belonging to
$H^s(\mathbb R^d)$ for $s<0$, and to $L^p(\mathbb R^d)$ for $1 \le p < 2$, but
not to $L^2(\mathbb R^d)$, which does not admit any local solution that can be
approximated by smooth solutions in a reasonable sense.
