I know it sounds absurd, but say we can move the point $X$ (the one we want to get the location of) and get as many measurements as we can to the other reference points. Is it theoretically possible to create a 3D model of the (approximate) relative positions of the reference points?
The input is a series of distance vectors and the output should be their positions in a freely chosen coordinate system.
Food for thought: Say in probe position $q_0$ (unknown) we get the distance vector $(2,3)$ to the set of reference points $(p_0,p_1)$ and in the probe position $q_1$ (also unknown) we get $(3,2)$. We can get close to no information, but we can at least deduce that we moved towards point $p_1$ and away from $p_0$, but maybe if we had more reference points we could get a better picture.
If it cannot be solvable, would it if we had distances and/or orientation changes between pairs of probe positions?
Edit: I modelled the problem in Octave by trying to solve an equation system of distances given that the unknowns are the reference and the probing locations. For this test I only modelled in 2d, guessing that a lower number of dimensions would lead to an easier problem.
I modelled 3 reference points with fixed locations, so that's 6 unknown variables for the solver (their X and Y coordinates). For each added probing location, I get 3 more distance equations for the equation system and 2 more unknowns. If I have 6 probing locations, my equation system has 3x6=18 equations and 2*(3+6)=18 unknowns. Assuming the equations are linearly independent (strong assumption!), in theory I could solve for all unknowns, I guess.
Result: No success. Either the equations are not linearly independent or my script was buggy, but I'm leaning towards the former.
If your points $p_0$ and $p_1$ are not defined, then you can't find out unknown position of $ q_0$ using only $d(q_0, p_0)$ and $d(q_0, p_1)$ because there will be infinite number of choices - here's how to imagine it:
You can pick out any $q_0$ with probability $1/|\mathbb R^3|$, then you can imagine 2 spheres around that $q_0$ with radiuses $p_0$ and $p_1$. And any combination of those radiuses will give you particular lengths chosen, say (2, 3) or (10, 20) or whatever.
You may consider this as 'approximation'.