What are compact dimensions in string theory? It is often said that string theory describes the world at the most fundamental level and is independent of the background, that is, not the strings are in space-time, but the space-time itself is emerging and emerges from the strings. Okay, but strings vibrate in 10 spacetime dimensions, 6 of which are folded and compact, have their own shape. What are these dimensions, do they have a physical meaning? If not, where do the strings vibrate?
 A: The spacetime of the string worldsheet is what is fundamental in perturbative string theory, not the target spacetime. You can define the worldsheet and its dynamics in an intrinsic way, without making reference to a "container space". In fact, that was a Gauss great achievement, "manifolds exist with independence of whether or not they accept embedings on bigger geometric spaces". That's the idea of intrinsic geometry.
After having defined the worldsheet and its dynamics (via the Polyakov action) you are in principle free to study the quantum dynamics of that theory. The miracle, is that the dynamics of the scalar fields of the string worldsheet behave as coordinates of a spacetime; more precisely, the moduli space of those scalars is a spacetime.
Even more spectacularly, the consistency of the quantum mechanics of those scalar fields imprints physical dynamics on that moduli space; the vanishing of the beta function (required to preserve the worldsheet conformal invariance) of the scalar fields is equivalent to Einstein's field theory equations on the target (why are there gravitons in string theory). Projecting tachyons of the worldsheet spectrum require the addition of spinor fields in the target, that ultimately can be seen as an explanation for the existence of matter in spacetime. The vanishing of a quantum anomaly of the Virasoro algebra fixes the dimension of spacetime (All of string theory's power, beauty depends on quantum mechanics) and many other physical masterpieces.
In summary: The fundamental objects of perturbative string theory are the fundamental strings. The spacetime existence is a derivable consequence of the quantum mechanics of the worldsheet CFT.
Further reading:

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*How and why strings generalize geometry.

*First-quantized formulation of string theory is healthy
A: the "compactified dimensions" exist on length scales so small that they are invisible to us, and each of them "points" in a "direction" which does not exist in the 3-dimensional space that you and I inhabit. For our purposes, they are mathematical abstractions which make consistent mathematical and logical sense but which have no aspects which we 3-dimensional organisms can easily visualize.
A: In topology, a compact space contains all its limit points. For example, the sequence 1,2,3... diverges on the real number line. But if you add a the number $\infty$, it converges to that point.
A circle is compact. There are no "holes" in the space like the missing point at infinity. So all sequences that get closer and closer to something converge on something.
The properties of a compact space are different from a non-compact space. For example, all continuous functions on a compact space are bounded. The function $y = 1/x$ is continuous and unbounded on the half open interval $(0,1]$. It is unbounded. But on the closed interval $[0,1]$, you can't define that function on the entire space.
In string theory, some dimensions are like the real number line, and some are like circles. The circular ones are so small that we don't see them. If you move a very tiny distance in a compact dimension, you are back where you started. You never get far enough from the start to measure.
