Can bodies of arbitrarily large density exist in general relativity? I've read that "in General Relativity there cannot
be bodies of an arbitrarily large mass with the given volume" [Matvei Petrovich Bronstein
and Soviet Theoretical
Physics in the Thirties, pg 106]. How to justify this? I found this article
https://link.springer.com/article/10.1007/BF00713098
where a spherically symmetric, perfect fluid solution's average density is shown to be bounded, but the same does not hold for a spheroid body... 
Can anyone provide a nice explanation to my question or further readings on the topic? Is it a general feature, or not, that in GR we cannot always pack a very massive body within a given volume?
 A: The Bonnor paper is from 1972, which is before people really took black holes seriously (the term "black hole" was just starting to gain currency) and not long after the 1965 Penrose singularity theorem. People still tended to have the attitude that singularities in general relativity were a mathematical artifact and would not occur in physically realistic solutions.
In terms of dimensional analysis, GR itself can't have an upper limit on density, because density isn't a unitless quantity in GR's system of geometrized units. The Bonnor paper seems to be mischaracterizing the Bondi result, or at the very least interpreting it in a way that people wouldn't normally describe it today. Known as the Buchdahl-Bondi limit, it's actually a limit on the dimensionless quantity $m/r$, not on the dimensionful density. The Buchdahl-Bondi limit is $2m/r < 8/9$.
The limit is not really a limit on what can exist according to GR, it's a limit on what can exist as a spherical body composed of a perfect fluid, surrounded by vacuum, in static equilibrium, with a zero cosmological constant. As an example of why we need so many conditions, FLRW cosmological models have an unbounded density near the big bang, but they don't demonstrate gravitational collapse or the formation of black holes.

I've read that "in General Relativity there cannot be bodies of an arbitrarily large mass with the given volume" [Matvei Petrovich Bronstein and Soviet Theoretical Physics in the Thirties, pg 106].

Note that this doesn't actually refer to density, and is true based on the Buchdahl-Bondi limit provided that the volume is spherical and that the other conditions above also hold.
From a modern perspective, a more important result than the Buchdahl-Bondi limit is the Penrose singularity theorem, which shows that when gravitational collapse proceeds beyond the formation of a trapped surface, there is guaranteed to be a singularity.
A: Good question -- no, bodies of arbitrarily high mass cannot exist in reality.
From the paper that you linked, Bonner stars, further reading here, are special known solutions where one considers a static spacetime, so no time evolution, and then excretes mass onto the star. It is a valid solution but for a specific set of conditions, namely exclusion of time evolution, and therefore are not physical.
In general, considering time evolution stars cannot be arbitrarily dense as high density for a fixed volume means arbitrarily high mass since
$$\rho=\frac{m}{V}.$$
With enough mass, the inward gravity becomes strong enough to over come the internal outward fusion pressure and the star will collapse into itself under its own weight. Usually this forms neutron stars or if the gravity is high enough, black holes.
