  We study the high-energy phase diagram of a two-dimensional
spin-$\frac{1}{2}$ Heisenberg model on a square lattice in the presence of
either quenched or quasiperiodic disorder. The use of large-scale tensor
network numerics allows us to compute the bipartite entanglement entropy for
systems of up to $60 \times 7$ lattice sites. We provide evidence for the
existence of a many-body localized regime for large disorder strength that
features an area law in excited states and that violates the eigenstate
thermalization hypothesis. From a finite-size analysis, we determine an
estimate for the critical disorder strength where the transition to the ergodic
regime occurs in the quenched case.
