Aims. We use stellar line-of-sight velocities to constrain the dark matter-density profile of Eridanus 2, an ultra-faint dwarf galaxy ($M_mathrm{V} = -7.1$, $M_* approx 9 times 10^4,M_odot$). We furthermore derive constraints on fundamental properties of self-interacting and fuzzy dark matter scenarios. Methods. We present new observations of Eridanus 2 from MUSE-Faint, a survey of ultra-faint dwarf galaxies with MUSE on the Very Large Telescope, and determine line-of-sight velocities for stars inside the half-light radius. Combined with literature data, we have 92 stellar tracers out to twice the half-light radius. We constrain models of cold dark matter, self-interacting dark matter, and fuzzy dark matter with these tracers, using CJAM and pyGravSphere for the dynamical analysis. Results. We find substantial evidence for cold dark matter over self-interacting dark matter and weak evidence for fuzzy dark matter over cold dark matter. We find a virial mass $M_{200} sim 10^8,M_odot$ and astrophysical factors $J(alpha_mathrm{c}^J) sim 10^{11},M_odot^2,mathrm{kpc}^{-5}$ and $D(alpha_mathrm{c}^D) sim 10^2$-$10^{2.5},M_odot,mathrm{kpc}^{-2}$. We do not resolve a core ($r_mathrm{c} < 47,mathrm{pc}$, 68-% level) or soliton ($r_mathrm{sol} < 7.2,mathrm{pc}$, 68-% level). These limits are equivalent to an effective self-interaction coefficient $fGamma < 2.2 times 10^{-29},mathrm{cm}^3,mathrm{s}^{-1},mathrm{eV}^{-1},c^2$ and a fuzzy-dark-matter particle mass $m_mathrm{a} > 4.0 times 10^{-20},mathrm{eV},c^{-2}$. The constraint on self-interaction is complementary to those from gamma-ray searches. The constraint on fuzzy-dark-matter particle mass is inconsistent with those obtained for larger dwarf galaxies, suggesting that the flattened density profiles of those galaxies are not caused by fuzzy dark matter. (Abridged)
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