Aims. It has been shown that the ultra-faint dwarf galaxy Eridanus 2 may host a stellar cluster in its centre. If this cluster is shown to exist, it can be used to set constraints on the mass and abundance of massive astrophysical compact halo objects (MACHOs) as a form of dark matter. Previous research has shown promising expectations in the mass range of 10−100 M⊙, but lacked spectroscopic measurements of the cluster. We aim to provide spectroscopic evidence regarding the nature of the putative star cluster in Eridanus 2 and to place constraints on 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 derived 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 km s−1 and 76.0+3.2−3.7 km s−1 for the cluster and the bulk of Eridanus 2, respectively, as well as intrinsic velocity dispersions of < 7.6 km s−1 (68% upper limit) and 10.3+3.9−3.2 km s−1, respectively. This indicates that the cluster most likely exists as a distinct dynamical population hosted by Eridanus 2 and that it does not have a 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 derived constraints on the fraction of dark matter that can consist of MACHOs with a given mass between 1 and 105 M⊙. For dark matter consisting purely of MACHOs, the mass of the MACHOs must be less than ∼7.6 M⊙ and ∼44 M⊙ at a 68- and 95% confidence level, respectively.Key words: dark matter / galaxies: individual: Eridanus 2 / galaxies: star clusters: individual: Eridanus 2 cluster / techniques: imaging spectroscopy⋆ Table 2 is only available at the CDS via anonymous ftp to cdsarc.u-strasbg.fr (130.79.128.5) or via http://cdsarc.u-strasbg.fr/viz-bin/cat/J/A+A/635/A107⋆⋆ Based on observations made with ESO Telescopes at the La Silla Paranal Observatory under programme ID 0100.D-0807.⋆⋆⋆ Current address: Kapteyn Astronomical Institute, University of Groningen, PO Box 800, 9700 AV Groningen, The Netherlands.