We discuss the stability of multiply strange baryonic systems, in the context of a mean field approach obtained from an underlying set of phenomenological meson-baryon interactions. The coupling parameters which determine the conventional {sigma} + {omega} mean fields (Hartree potentials) seen by varius baryon species (N, {lambda}, {Xi}) in the many-body system are constrained by reproducing the trend of observed binding energies of single particle (N, {lambda}, {Xi}) states, as well as the energy per particle and density of non- strange nuclear matter. We also consider additional scalar ({sigma}{asterisk}) and vector ({phi}) fields which couple strongly to strange baryons. The couplings of these fields are adjusted to produce strong hyperon-hyperon interactions, as suggested by the data on {Lambda}{Lambda} hypernuclei. Extrapolating this approach to systems of large strangeness S, we find a broad class of objects composed of neutrons, protons, {Lambda}`s and {Xi}`s, which are stable against strong decay. In these systems, the presence of filled {Lambda} orbitals blocks the strong decay {Xi} N {yields} {Lambda}{Lambda}, leading to a strangeness fraction {integral}{sub s} = {vert_bar}S{vert_bar}/A {approx} 1, density p {approx} (2{minus}3){sub po}, and charge fraction {integral}{sub q} in the range {minus}0.1 < q/A < 0.1, comparable to that of hypothetical stable strange quark matter (``strangelets``), but with a low binding energy per particle E{sub B}/A{approx}{minus}10 to {minus}20 MeV. Such weakly bound multi-strange objects can be stable for very large A, unlike ordinary nuclei, since the Coulomb repulsion generated by the protons is largely cancelled by the presence of a comparable number of {Xi}`s, leading to a small net charge (positive or negative) of order A{sup 1/3}.