  Due to $e^+e^-$-pair production in the field of supercritical $(Z \gg
Z_{cr}\approx 170 $) nucleus an electron shell, created out of the vacuum, is
formed. The distribution of the vacuum charge in this shell has been determined
for super-charged nuclei $Ze^3 \ga 1$ within the framework of the Thomas-Fermi
equation generalized to the relativistic case. For $Ze^3 \gg 1$ the electron
shell penetrates inside the nucleus and almost completely screens its charge.
Inside such nucleus the potential takes a constant value equal to $V_0=-(3\pi^2
n_p)^{1/3} \sim -2m_{\pi}c^2$, and super-charged nucleus represents an
electrically neutral plasma consisting of $e,p$ and $n$. Near the edge of the
nucleus a transition layer exists with a width $\lambda \approx \alpha^{-1/2}
\hbar/m_{\pi} c\sim 15$ fm, which is independent of $Z (\hbar/m_{\pi} c \ll
\lambda \ll \hbar/m_e c)$. The electric field and surface charge are
concentrated in this layer. These results, obtained earlier for hypothetical
superheavy nuclei with $Z \sim A/2\la 10^4 \div 10^6$, are extrapolated to
massive nuclear density cores having a mass number $A \approx
(m_{Planck}/m_n)\sim 10^{57}$. The problem of the gravitational and
electrodynamical stability of such objects is considered. It is shown that for
$A \ga 0.04 (Z/A)^{1/2}(m_{Planck}/m_n)^3$ the Coulomb repulsion of protons,
screened by relativistic electrons, can be balanced by gravitational forces.
The overcritical electric fields $E\sim m^2_{\pi} c^3/e\hbar$ are present in
the narrow transition layer near the core surface.
