Extra dimensions in general have to be compact, since a four-dimensional description fits the world we perceive and measure (so far) very well. It is this compactness that sets a length scale. Usually, we assume that the four "regular" space-time dimensions are not compact, i.e. extended infinitely. In higher dimensions concepts like angular momentum, including Spin, work differently.

The size of this length scale is not set a priori, and a mechanism giving a precise prediction for the size of the compact directions would be a major breakthrough. However, a number of effects in the low-energy description of a theory with extra dimensions depend crucially on this scale - most prominently the spacing of the Kaluza-Klein mass tower, but also e.g. the scale of symmetry breaking in GUT models with extra dimensions. How available these effects are to experiments is highly model-dependent, though. For example, if one introduces brane-localized fields, the corresponding particles do not show a Kaluza-Klein tower of excitations.

The size and other properties of the extra dimensions become a dynamical quantity if one incorporates General Relativity, which String Theory does inherently. Since these so-called moduli degrees of freedom are usually unstable, a realistic theory must incorporate some mechanism of moduli-stabilization. There are a number of mechanisms on the market, each with benefits and detriments of its own.

So, in conclusion

1. We do not observe the higher Kaluza-Klein states of any particle so far. Even with fully brane-localized matter, the graviton should have KK states and their non-obervation puts an upper limit on the size of extra dimensions that shrinks with higher energies probed. A particularly well-motivated scale is the GUT scale, where a grand unified theory might be broken in the process of compactification.
2. The "regular" space-time dimensions are considered infinite. It is not fully excluded that we do live in a compact universe of immense radius, though. These theories are not very popular, since a compact universe immediately implies Lorentz violation (most notably a minimum energy for free photons) and the Lorentz symmetry of our universe is extremely well-tested!

Note that nothing I wrote depends on the actual UV completion, i.e. my statements are independent of String theory.

Extra dimensions in general have to be compact, since a four-dimensional description fits the world we perceive and measure (so far) very well. It is this compactness that sets a length scale. Usually, we assume that the four "regular" space-time dimensions are not compact, i.e. extended infinitely. In higher dimensions concepts like angular momentum, including Spin, work differently.

The size of this length scale is not set a priori, and a mechanism giving a precise prediction for the size of the compact directions would be a major breakthrough. However, a number of effects in the low-energy description of a theory with extra dimensions depend crucially on this scale - most prominently the spacing of the Kaluza-Klein mass tower, but also e.g. the scale of symmetry breaking in GUT models with extra dimensions. How available these effects are to experiments is highly model-dependent, though. For example, if one introduces brane-localized fields, the corresponding particles do not show a Kaluza-Klein tower of excitations. Therefore, any limits on "the size of extra dimensions" have to be weighted carefully with the model assumptions.

The size and other properties of the extra dimensions become a dynamical quantity if one incorporates General Relativity, which String Theory does inherently. Since these so-called moduli degrees of freedom are usually unstable, a realistic theory must incorporate some mechanism of moduli-stabilization. There are a number of mechanisms on the market, each with benefits and detriments of its own.

So, in conclusion

1. We do not observe the higher Kaluza-Klein states of any particle so far. Even with fully brane-localized matter, the graviton should have KK states and their non-obervation puts an upper limit on the size of extra dimensions that shrinks with higher energies probed. A particularly well-motivated scale is the GUT scale, where a grand unified theory might be broken in the process of compactification.
2. The "regular" space-time dimensions are considered infinite. It is not fully excluded that we do live in a compact universe of immense radius, though. These theories are not very popular, since a compact universe immediately implies Lorentz violation (most notably a minimum energy for free photons) and the Lorentz symmetry of our universe is extremely well-tested!

Note that nothing I wrote depends on the actual UV completion, i.e. my statements are independent of String theory.