  We report the magnetization ($\chi$, $M$), specific heat ($C_{\text{P}}$),
and neutron powder diffraction results on a quasi-two-dimensional $S$ = 2
square lattice antiferromagnet Ba$_2$FeSi$_2$O$_7$ consisting of FeO$_4$
tetragons with a large compressive distortion (27%). Despite of the
quasi-two-dimensional lattice structure, both $\chi$ and $C_{\text{P}}$ present
three dimensional magnetic long-range order below the N\'eel temperature
$T_{\text{N}}$ = 5.2 K. Neutron diffraction data shows a collinear $Q_{m}$ =
(1,0,0.5) antiferromagnetic (AFM) structure with the in-plane ordered magnetic
moment suppressed by 26% below $T_{\text{N}}$. Both the AFM structure and the
suppressed moments are well explained by the Monte Carlo simulation with a
large single-ion ab-plane anisotropy $D$ = 1.4 meV and a rather small in-plane
Heisenberg exchange $J_{\text{intra}}$ = 0.15 meV. The characteristic two
dimensional spin fluctuations can be recognized in the magnetic entropy release
and diffuse scattering above $T_{\text{N}}$. This new quasi-2D magnetic system
also displays unusual non-monotonic dependence of the $T_{\text{N}}$ as a
function of magnetic field $H$.
