The question I have arises from taking seriously the idea that, in a physical theory, one should first define all physical quantities via (thought) experiment.
Justification and motivation for this view
What follows are a couple of reasons why I think this is a reasonable view to take. (In particular, if you need no motivation, then you can safely skip this section.)
The first example that that comes to mind is the way in which some of the fundamentals of special relativity can be derived from the idea that basic concepts like "length" and "time" are not 'god-given' but instead are defined by an experiment.
In a similar way, in quantum field theory, if one first requires that we define the coupling constant of an interaction via a scattering experiment, then, not only is coupling constant renormalization no longer an ad hoc trick to remove infinities, but it is a completely natural thing to do, and furthermore, the "running of the coupling constant" appears in such a way that almost makes it obvious that this should be the case (as the experiment used to define the coupling constant depends on the energy of the particles, of course the coupling constant is going to depend on the energy scale!).
This philosophy also makes the idea that physics is inherently probabilistic quite natural: if a physical quantity is defined by performing experiments, then unless every experiment resulted in the same measured number (they do not), then the physical quantity will itself not be a single number but instead a probability distribution (defined by a sequence of measurements).
I won't be so bold as to claim that one cannot make a counter-argument to the claim that defining physical quantities via (thought) experiment is crucial to any physical theory, but hopefully these examples make it at least a reasonable thing to try to do. In any case, in order to understand the motivation for the following question, you will have to grant me for the time being that such a philosophy is reasonable.
This idea applied to space-time
If you do take this idea seriously, then in any physical theory in which space-time is modeled as a manifold, an observer should be able to perform experiments which associate to each point in space-time a tuple of numbers, the coordinates of that point with respect to that observer. (I am being a bit sloppy: of course, the observer can make different choices as to how they perform their measurements (e.g. by using different axes) which will result in a different set of coordinates.) This works fine classically, and seems to be completely compatible with general relativity. If we want to incorporate quantum mechanics, however, we immediately run into a problem: the results of our experiments are not just numbers (with units), but numbers together with an uncertainty, which, from the perspective of quantum mechanics, is not just a result of us being poor experimental physicists, but are absolutely fundamental to the theory and cannot be eliminated. Thus, it seems that if we seek to incorporate quantum mechanics, 'points' in space-time become 'fuzzy'. This suggests to me that any theory which seeks to model 'space-time' and also incorporate principles of quantum mechanics cannot model space-time as a manifold as would be done in classical theories like general relativity.
String theory of course does seek to incorporate both space-time and quantum mechanical principles into the theory, but from the very beginning the theory seems to assume a classical model of space-time (I have in mind both the Nambu-Goto and Polyakov action, which both assume a priori that the target space-time is a classical (riemannian) manifold). When first studying the subject, I supposed I had always assumed that eventually, once more of the theory was developed, we would come back and explain why this classical model of space-time was a reasonable thing to do, but after almost two years worth of courses on the subject, this point was never addressed. Thus, I ask: What is string theory's justification for a priori modeling space-time as a manifold?
Finally, I don't mean to pick on string theory in particular. I would have the same question for pretty much any theory of quantum gravity; however, as string theory is really the only one I know anything about, I cannot as confidently say that other approaches don't address this issue.