  We aim to provide spectroscopic evidence regarding the nature of the putative
star cluster in Eridanus 2 and to place constraints on the mass and abundance
of massive astrophysical compact halo objects (MACHOs) as a constituent of dark
matter. Methods. We present spectroscopic observations of the central square
arcminute of Eridanus 2 from MUSE-Faint, a survey of ultra-faint dwarf galaxies
with the Multi Unit Spectroscopic Explorer on the Very Large Telescope. We
derive line-of-sight velocities for possible member stars of the putative
cluster and for stars in the centre of Eridanus 2. We discuss the existence of
the cluster and determine new constraints for MACHOs using the Fokker-Planck
diffusion approximation. Results. Out of 182 extracted spectra, we identify 26
member stars of Eridanus 2, seven of which are possible cluster members. We
find intrinsic mean line-of-sight velocities of
$79.7^{+3.1}_{-3.8}\,\mathrm{km}\,\mathrm{s}^{-1}$ and
$76.0^{+3.2}_{-3.7}\,\mathrm{km}\,\mathrm{s}^{-1}$ for the cluster and the bulk
of Eridanus 2, respectively, and intrinsic velocity dispersions of
${<}7.6\,\mathrm{km}\,\mathrm{s}^{-1}$ (68-$\%$ upper limit) and
$10.3^{+3.9}_{-3.2}\,\mathrm{km}\,\mathrm{s}^{-1}$, respectively. This
indicates the cluster most likely exists as a distinct dynamical population
hosted by Eridanus 2, without surplus of dark matter over the background
distribution. Among the member stars in the bulk of Eridanus 2, we find
possible carbon stars, alluding to the existence of an intermediate-age
population. We derive constraints on the fraction of dark matter that can
consist of MACHOs with a given mass between $1$-$10^5\,M_\mathrm{sun}$. For
dark matter consisting purely of MACHOs, the mass of the MACHOs must be less
than ${\sim}7.6\,M_\mathrm{sun}$ and ${\sim}44\,M_\mathrm{sun}$ at a $68$- and
$95$-$\%$ confidence level, respectively. (Abridged)
