Bound states
In retrospect the below answer is a bit long winded, so I will summarize the basic point: to get "orbital" motion, one simply needs that some object is trapped in the potential well of another, meaning it doesn't have enough kinetic energy to escape. If the potential well is rotationally symmetric then there will be conservation of angular momentum, and if the potential has $1/r$ dependence then the orbits will be Keplerian (at least classically), but that is not necessarily important. All that matters is that you can have a hierarchy: some objects A, B, C, . . . with decreasing "charges" (e.g. gravitational mass, or electric charge) $m_A\gg m_B \gg m_C$, . . . with separation distances $r_{AB}\gg r_{BC}\gg$ . . . so that B is trapped in the potential well of A, C is trapped in the potential well of B, etc. In the example below, I chose A=galactic core, B=a star, C=a planet, D=a moon (and a satellite can orbit the moon, and in principle even something very tiny could orbit the satellite, but it would be so weakly bound that tiny perturbations would give it enough energy to escape).
The quantum mechanical version of what I am describing is a bound state. Quarks are bound (by the strong force) into hadrons (protons and neutrons), which are bound (again by the strong force) into atomic nuclei, which form bound states with electrons (via the electromagnetic force) to make atoms, which can further form bound states (by chemical bonding, which is electromagnetic in origin) to make molecules, and so on until you get to sizes where electric charge is screened and Newtonian gravity takes over as described in the paragraph above and the original answer below.
Original answer below
There are many potential directions in which your question could be interpreted and answered. One, which comes from a physics perspective, is symmetry. In this case, rotational symmetry. The laws of physics are rotationally invariant, and as a consequence of Noether's theorem, angular momentum is a conserved quantity.
In orbital mechanics specifically, one generally is dealing with central potentials, e.g. the potential energy of a massive object moving around the sun only depends on the distance from the sun (and decays is one over distance). Since this potential does not depend on the angle one makes with any particular axis through the sun, it is rotationally symmetric and thus angular momentum is conserved. (A note: although it is true that electrons do not "orbit" atomic nuclei, they still have conserved angular momentum!)
Now to the different scales portion of the question. At the scale of planetary motion, gravity is the only important force to consider, all other forces are essentially completely screened at this scale (because the relevant objects, e.g. asteroids or planets, are charge-neutral). The motion of the objects in the solar system relative to each other can, to a good approximation, be approximated by only considering the gravitational force of the sun, as described in the previous paragraph. Inter-object forces (e.g. gravitational attraction between planets) can be treated perturbatively.
Similarly at other scales, e.g. a single planet gravitationally interacting with its moons can be approximated to a good approximation as a collection of two-body problems. Each object orbits in the near-spherically-symmetric potential well of the nearest object which has a mass orders of magnitude larger than itself, and the gravitational attraction of smaller nearby objects act only perturbatively relative to the dominant nearby attractive body.
Let me try to be slightly more precise about that. Let's start at a large scale: the solar system orbits around the galactic core. For simplicity let's model the galactic core as a point mass.$^\dagger$ The solar system is about $10^{20}$ meters from the galactic core, and has a radius of about $10^{10}$ meters. The size of the solar system relative to the size of the galaxy is about the same as the size of an atom relative to the size of your body, so we can treat it as a point mass. Treating this as a 2-body problem, the gravitational potential that the solar system orbits in might look something like this:
where $a_{\mathrm{galactic}}$ is a lengthscale on the order $10^{20}$ meters. But of course, the solar system itself is made of a bunch of small massive objects, the largest of the which is the sun, and so itself has a gravitational potential. If I greatly exaggerate that, and throw the Earth in (I must stress this is not to scale at all, only for illustrative purposes), the potential looks something like this:
We see that while the solar system moves in the potential well of the galactic core, the Earth also moves in the potential well of the sun. Of course, the Earth itself also has a potential, so we could further break that down (again, only illustrative, and throwing the moon in too) as
So the lesson is this: each object moves in the local potential well of the nearest massive gravitational source. Each object then creates a smaller local potential well around itself that smaller objects can orbit in, and so on. In fact, if we were to correct the above plot to accurately indicate the distances and masses involved, we would see that at each of these scales the $1/r$ potential of the next larger object is almost completely flat. That is to say, for example $1/(a_{\mathrm{galactic}}+a_{\mathrm{solar\,system}}) \approx 1/(a_{\mathrm{galactic}})$ to one part in $10^{10}$. This means that every object in the solar system feels essentially the same gravitational force from the galactic core, regardless of where in the solar system they are, because the galactic core is, practically, infinitely far away, and its gravitational force points uniformly in the same direction at each point within the solar system. Similarly the sun is very far away from the Earth and moon, relative to their separation distance. All that is to say that tidal forces are negligible, and these systems can be treated to a very good approximation as two-body orbital pairs, with their center of mass subject to an external uniform force.
${}^\dagger$ this approximation is not for realistic modelling, just to make a point, see the comments.
