So, planetary orbits are unstable under an inverse cubic force--that's old news.

But... that's only the external field of a (hyper)spherically symmetric charge distribution. Inside a uniform charge distribution which obeys Gauss's law, the force is linear, and the potential is that of a harmonic oscillator, in any number of dimensions.

Given a truly uniform charge distribution, the orbits of test particles would be ellipses, but centered on the geometric center rather than one focus of the ellipse. This could stabilize solar systems if there is some sort of dark matter which can form uniform hyperballs of gravitational charge which regular matter can pass through unimpeded... but that's a little excessively contrived. It's kind of neat that the aspect ratio of the ellipse is completely free, though, so you can get arbitrarily extreme orbital seasons, which repeat twice per sidereal year.

What's not contrived, though, is an assumption of uniform charge distribution within nuclear matter. That means that, when the non-zero physical size of an atomic nucleus is taken into account, the inverse-cubic electric potential which leads to unstable solutions to the Schroedinger equation for 4D hydrogen, gets patched with a section of harmonic oscillator which eliminates the singularity and bounds the potential energy from below.

Also not contrived are galaxies and globular clusters. As a first-order approximation, a perfect uniform hyperspherical distribution of stars would have an effective harmonic potential in its interior, such that stars would stably orbit. So, in practice, no galaxy or cluster would remain perfectly uniform; stars would move more quickly through the center, meaning less residence time, meaning less density; so, a stable galaxy or globular cluster could end up with a a low-density interior and a higher density periphery. But, that's not a problem--it just means that the force grows faster than linearly with radius, which means orbits are weird rosettes rather than closed ellipses, but they are still bound and stable. Supposing we don't go with contrived dark-matter stabilized solar systems, habitable planets could perhaps exist inside globular clusters with a suitable stellar density to produce a comfortable average level of interior illumination--and since a shell of Gaussian charge (with the charges in this case being light sources) produce a constant interior potential in any number of dimensions, a planet inside a cluster with a high surface concentration of stars could have a quite chaotic path while still receiving a near-constant level of illumination across the entire surface at all times!