  In this paper the spectral properties of Dirac operators $A_\eta$ with
electrostatic $\delta$-shell interactions of constant strength $\eta$ supported
on compact smooth surfaces in $\mathbb{R}^3$ are studied. Making use of
boundary triple techniques a Krein type resolvent formula and a
Birman-Schwinger principle are obtained. With the help of these tools some
spectral, scattering, and asymptotic properties of $A_\eta$ are investigated.
In particular, it turns out that the discrete spectrum of $A_\eta$ inside the
gap of the essential spectrum is finite, the difference of the third powers of
the resolvents of $A_\eta$ and the free Dirac operator $A_0$ is trace class,
and in the nonrelativistic limit $A_\eta$ converges in the norm resolvent sense
to a Schr\"odinger operator with an electric $\delta$-potential of strength
$\eta$.
