To try and explain this, since you are mostly interested in the distances between atoms, let me show a picture of the pair distribution function for the Lennard-Jones liquid, at a fairly low temperature (the black curve). This was calculated from a molecular dynamics (computer) simulation.
With the origin of coordinates on an (arbitrary) atom, $g(r)$ shows the probability of finding another atom at a distance $r$, divided by the same quantity for an ideal gas having the same density $\rho$. So the combination $\rho g(r)$ is a "local density" of other atoms, around a typical atom. You'll notice a sharp peak at $r\approx\sigma$ (where $\sigma$ is the atomic diameter) which indicates a fairly well defined shell of nearest neighbours. Then there are subsidiary peaks at $r\approx 2\sigma$, $3\sigma$ etc. This is somewhat similar to what you see in a solid: more-or-less well defined next-nearest-neighbour, and next-next-nearest-neighbour distances. At large distance, though, the function settles down to a constant value: there are no long-range positional correlations in a liquid. In a solid, the peaks would be sharper, the structure more complicated, and the long distance correlations would not disappear; also the function would depend on direction in a solid, whereas it is isotropic in a liquid. So, there are neighbours with (roughly) the same nearest-neighbour distance as in a solid, there is also some local structure in the liquid as you look further away from a given atom, and this (together with the high overall density $\rho$, which for liquids is quite similar to solids) all contributes to making liquids fairly incompressible, similar to solids.
For comparison, I've shown in red the same quantity for a low-density Lennard-Jones gas, at the same temperature. There is a peak around $r\approx\sigma$ which is due to the attractive part of the Lennard-Jones potential. It's a bit higher, and a bit broader, than in the liquid. However, this is (again) just relative to the ideal gas at the same density. The actual density of neighbouring atoms, in a gas, is low: most atoms are not near other atoms. And also, there are no subsidiary peaks.
Don't forget that a liquid can be continuously converted into a gas, by raising the temperature above the critical point, then reducing the pressure, then (if you like) reducing the temperature again. We identify them as two separate phases because it is also possible to see a sharp phase transition between them, if we try to do this interconversion below the critical point. So there will always be a region of the phase diagram in which the properties (including the compressibility) are intermediate between the values we associate with liquids/solids and gases.
EDIT following OP change to question, and OP comments.
Other atomic liquids have fairly similar-looking plots of $g(r)$. For solids, I roughly described the differences in the appearance of $g(r)$ in my answer. I don't have a plot for a solid immediately to hand.
There are two main points here. Firstly, compressing a solid, or a liquid, will push the atoms, which are already close, even closer. This is resisted by the repulsive part of the interatomic potential: the energy goes up rapidly as the atoms begin to overlap. Hence the bulk modulus of a solid or a liquid is quite high, and we loosely speak of these phases as being "incompressible".
The second point is that the atoms in a liquid are continuously in motion, at thermal equilibrium. It is still perfectly possible to define a distribution of distances, as an average, and this is what is represented by $g(r)$. Quantities such as the pressure, and the bulk modulus, can be calculated as integrals over $r$, involving $g(r)$ and the interaction potentials (such as Lennard-Jones). The atomic motion is averaged over, in order to get $g(r)$. A computer simulation (such as molecular dynamics, in which the atoms actually move around following Newton's equations of motion) can be used to calculate $g(r)$ for a liquid; there are also some fairly complicated theories for this function.
You seem to be starting from an assumption that there is a particular configuration of atoms with well defined distances, and then you seem to be worrying about the fact that they are not fixed. When considering a liquid, we cannot assume that the atoms are fixed; but also we cannot ignore their interactions. This is why liquids are interesting, and why simple models (such as a harmonic solid, or an ideal gas) do not apply.