  We explore static and spherically symmetric solutions of the 4-dimensional
semiclassical Einstein's equations using the quantum vacuum polarization of a
conformal field as a source. These solutions may be of interest for \black{the
study of} exotic, compact objects (ECOs). The full backreaction problem is
addressed by solving the semiclassical Tolman-Oppenheimer-Volkoff (TOV)
equations making use of effective equations of state inspired \black{by} the
trace anomaly and an extra simplifying and reasonable assumption. We combine
analytical and numerical techniques to solve the resulting differential
equations, both perturbatively and nonperturbatively in $\hbar$. In all cases
the solution is similar to the Schwarzschild metric up to the vicinity of the
classical horizon $r=2M$. However, at $r=2M + \varepsilon$, with
$\varepsilon\sim O(\sqrt{\hbar})$, we find a coordinate singularity. In
\black{the} case of matching with a static star, this leads to an upper bound
in the compactness, and sets a constraint on the family of stable ECOs. We also
study the corrections that the quantum-vacuum polarization induces on the
propagation of waves, and discuss the implications. For the pure vacuum case,
we can further extend the solution by using appropriate coordinates until we
reach another singular point, where this time a null curvature singularity
arises and prevents extending beyond. This picture qualitatively agrees with
the results obtained in the effective two-dimensional approach, and reinforces
the latter as a reasonable method.

General Relativity and Quantum Cosmology;High Energy Physics - Theory