There are exactly 2 force laws that can produce stable closed orbits: proportional (spring force) and inverse-square. The inverse square law occurs naturally for monopole central forces in 3 dimensions, while there isn't any space which naturally produces distance-proportional force fields, which is one of the arguments for why 3D space is special.
However, in space of any dimensionality, a sphere of uniform density composed of monopole field sources will produce an effective force which rises proportional to radius in the interior, before falling off on the exterior. That's our spring force law! So, if the sphere is composed of dark matter which doesn't interact other than gravitationally with other stuff passing through it, we can stick a sun and planets in there and get a nice stable solar-system-in-a-ball in any kind of space (well, any space with dimensionality > 1, anyway).
Now, we can get bound, but not necessarily closed, orbits with much less precision. E.g., bound-but-non-closed orbits happen naturally in 2D space, and radially non-uniform spherical distribution of dark matter could produce all sorts of effective force laws which fall off less quickly than $r^{-3}$ (the necessary condition for boundedness). But if we want closed orbits, the density profile of the dark matter sphere has to be absolutely uniform.
So, the question becomes "What properties must a dark matter particle, or particles, have to accumulate into solar-system-scale self-gravitating spheres while maintaining as-close-to-constant-as-possible density in the interior?"
Any state of matter is permissible--solid, liquid, or something more exotic. A perfect solution may not be possible, as perfectly rigid, perfectly incompressible matter is not possible in a relativistic universe, but in that case I am interested in just how good it can get.
For reference, something like "dark neutronium" would not work--the density of neutron star matter can vary over several orders of magnitude between the core and surface regions, and we don't want that much density anyway.