  The equation of state and phase diagram of strongly interacting matter
composed of $\alpha$ particles are studied in the mean-field approximation. The
particle interactions are included via a Skyrme-like mean field, containing
both attractive and repulsive terms. The model parameters are found by fitting
known values of binding energy and baryon density in the ground state of
$\alpha$ matter, obtained from microscopic calculations by Clark and Wang.
Thermodynamic quantities of $\alpha$ matter are calculated in the broad domains
of temperature and baryon density, which can be reached in heavy-ion collisions
at intermediate energies. The model predicts both first-order liquid-gas phase
transition and Bose-Einstein condensation of $\alpha$ particles. We present the
profiles of scaled variance, sound velocity and isochoric heat capacity along
the isentropic trajectories of $\alpha$ matter. Strong density fluctuations are
predicted in the vicinity of the critical point at temperature $T_c\approx
14~\textrm{MeV}$ and density $n_c\approx 0.012~\textrm{fm}^{-3}$.
