  In this paper we present a deep X-ray observation of the nearby M dwarf GJ
357 and use it to put constraints on the atmospheric evolution of its planet,
GJ 357 b. We also analyse the systematic errors in the stellar parameters of GJ
357 in order to see how they affect the perceived planetary properties. We
estimate the age of GJ 357 b by comparing the observed X-ray luminosity of its
host star, derived from a recent {\em XMM-Newton} observation {($\log{L_{\rm
x}}\,{\rm [erg/s]} = 25.73$), with $L_{\rm x} -$ age relations for M dwarfs. We
find that GJ 357 presents one of the lowest X-ray activity levels ever measured
for an M dwarf, and we put a lower limit on its age of $5$\,Gyr.} Using this
age limit, we perform a backwards reconstruction of the original primordial
atmospheric reservoir. Furthermore, by considering the systematic errors in the
stellar parameters, we find a range of possible planetary masses, radii, and
densities. From the backwards reconstruction of GJ 357 b's irradiation history
we find that the upper limit of its initial primordial atmospheric mass is
$\sim \rm 38M_{\oplus}$. An initial atmospheric reservoir significantly larger
than this may have survived through the X-ray and ultraviolet irradiation
history, hence being inconsistent with current observations that suggest a
telluric composition. In spite of the unlikelihood of a currently existing
primordial envelope, volcanism and outgassing may have contributed to a
secondary atmosphere. Under this assumption, we present three different
synthetic infrared spectra for GJ 357 b that one might expect, consisting of
$100\%~\rm CO_{2}$, $100\%~\rm SO_{2}$, and $75\%~ \rm N_{2}$, $24\%~\rm
CO_{2}$ and $1\%~\rm H_{2}O$.
