  How can a Schwarzschild-sized matter system avoid a fate of gravitational
collapse? To address this question, we critically reexamine the arguments that
led to the "Buchdahl bound", which implies that the minimal size of a stable,
compact object must be larger than nine eighths of its own Schwarzschild
radius. Following Mazur and Mottola, and in line with other counterexamples to
the singularity theorems, we identify large negative radial pressure extending
to the gravitational radius as the essential ingredient for evading the
Buchdahl bound. Our results are novel although consistent with many other
investigations of models of non-singular black holes. It is shown in particular
that a large negative pressure in the framework of classical GR translates into
a large positive pressure once quantum physics is incorporated. In this way, a
Schwarzschild-sized bound state of closed, interacting fundamental strings in
its high-temperature Hagedorn phase can appear to have negative pressure and
thus the ability to resist gravitational collapse.
