  We present the Chandra imaging and spectral analysis of NGC 4968, a nearby (z
= 0.00986) Seyfert 2 galaxy. We discover extended ($\sim$1 kpc) X-ray emission
in the soft band (0.5 - 2 keV) that is neither coincident with the narrow line
region nor the extended radio emission. Based on spectral modeling, it is
linked to on-going star formation ($\sim$2.6-4 M$_{\sun}$ yr$^{-1}$). The soft
emission at circumnuclear scales (inner $\sim$400 pc) originates from hot gas,
with kT $\sim$ 0.7 keV, while the most extended thermal emission is cooler (kT
$\sim$ 0.3 keV). We refine previous measurements of the extreme Fe K$\alpha$
equivalent width in this source (EW = 2.5$^{+2.6}_{-1.0}$ keV), which suggests
the central engine is completely embedded within Compton-thick levels of
obscuration. Using physically motivated models fit to the Chandra spectrum, we
derive a Compton-thick column density ($N_{\rm H} > 1.25\times10^{24}$
cm$^{-2}$) and an intrinsic hard (2-10 keV) X-ray luminosity of
$\sim$3-8$\times 10^{42}$ erg s$^{-1}$ (depending on the presumed geometry of
the obscurer), which is over two orders of magnitude larger than that observed.
The large Fe K$\alpha$ EW suggests a spherical covering geometry, which could
be confirmed with X-ray measurements above 10 keV. NGC 4968 is similar to other
active galaxies that exhibit extreme Fe K$\alpha$ EWs (i.e., $>$2 keV) in that
they also contain ongoing star formation. This work supports the idea that gas
associated with nuclear star formation may increase the covering factor of the
enshrouding gas and play a role in obscuring AGN.
