  We consider a paradigmatic solvable model of topological order in two
dimensions, Kitaev's honeycomb Hamiltonian, and turn it into a measurement-only
dynamics consisting of stochastic measurements of two-qubit bond operators. We
find an entanglement phase diagram that resembles that of the Hamiltonian
problem in some ways, while being qualitatively different in others. When one
type of bond is dominantly measured, we find area-law entangled phases that
protect two topological qubits (on a torus) for a time exponential in system
size. This generalizes the recently-proposed idea of Floquet codes, where
logical qubits are dynamically generated by a time-periodic measurement
schedule, to a stochastic setting. When all types of bonds are measured with
comparable frequency, we find a critical phase with a logarithmic violation of
the area-law, which sharply distinguishes it from its Hamiltonian counterpart.
The critical phase has the same set of topological qubits, as diagnosed by the
tripartite mutual information, but protects them only for a time polynomial in
system size. Furthermore, we observe an unusual behavior for the dynamical
purification of mixed states, characterized at late times by the dynamical
exponent $z = 1/2$ -- a super-ballistic dynamics made possible by measurements.
