  BaFe$_2$S$_3$ is a quasi-one-dimensional antiferromagnetic insulator that
becomes superconducting under hydrostatic pressure. The magnetic ordering
temperature, $T_N$, as well as the presence of superconductivity have been
found to be sample dependent. It has been argued that the Fe content may play a
decisive role, with the use of 5%mol excess Fe being reportedly required during
the synthesis to optimize the magnetic ordering temperature and the
superconducting properties. However, it is yet unclear whether an Fe
off-stoichiometry is actually present in the samples, and how it affects the
structural, magnetic and transport properties. Here, we present a systematic
study of compositional, structural and physical properties of
BaFe$_{2+\delta}$S$_3$ as a function of the nominal Fe excess $\delta$. As
$\delta$ increases, we observe the presence of an increasing fraction of
secondary phases but no systematic change in the composition or crystal
structure of the main phase. Magnetic susceptibility curves are influenced by
the presence of magnetic secondary phases. The previously reported maximum of
$T_N$ at $\delta$=0.1 was not confirmed. Samples with nominal $\delta$=0
present the lowest $T_N$ and the resistivity anomaly at the highest temperature
$T^*$ while, for $\delta \geq 0.05$, both quantities and the transport gap are
seemingly $\delta$-independent. Finally, we show that crystals free of
ferromagnetic spurious phases can be obtained by remelting samples with nominal
$\delta$=0.05 in a Bridgman process.
