  [abridged] We propose a model of the Universe (dubbed $\eta$CDM) featuring a
stochastic evolution of the cosmological quantities, that is meant to render
small deviations from homogeneity/isotropy on scales of $30-50\, h^{-1}$ Mpc at
late cosmic times, associated to the emergence of the cosmic web. Specifically,
we prescribe that the behavior of the matter/radiation energy densities in
different patches of the Universe with such a size can be effectively described
by a stochastic version of the mass-energy evolution equation. The latter
includes an appropriate noise term that statistically accounts for local
fluctuations due to inhomogeneities, anisotropic stresses and matter flows. The
evolution of the different patches as a function of cosmic time is rendered via
the diverse realizations of the noise term; meanwhile, at any given cosmic
time, sampling the ensemble of patches will originate a nontrivial spatial
distribution of the cosmological quantities. The overall behavior of the
Universe will be obtained by averaging over the patch ensemble. We assume a
physically reasonable parameterization of the noise term, gauging it against a
wealth of cosmological datasets. We find that, with respect to standard
$\Lambda$CDM, the ensemble-averaged cosmic dynamics in the $\eta$CDM model is
substantially altered in three main respects: (i) an accelerated expansion is
enforced at late cosmic times without the need for any additional exotic
component (e.g., dark energy); (ii) the spatial curvature can stay small even
in a low-density Universe; (iii) matter can acquire an effective negative
pressure at late times. We provide predictions for the variance of the
cosmological quantities among different patches of the Universe at late cosmic
times. Finally, we show that in $\eta$CDM the Hubble-tension is solved, and the
cosmic coincidence problem is relieved without invoking the anthropic
principle.
