In string theory, the idea is some dimensions are curled up and therefore "tiny". — Benkei
The only curling up that is done in string theory is mathematical. Like in Kaluza Klein theory a 4th dimension was introduced for the EM interaction.
A 5d metric (one for time) was introduced of which extra 8 components, 4 independent ones, on top of the 6 in the 4d metric (ignoring diagonal components).
These are supposed to induce a metric on a small curled up circle, present at every point in spacetime, perpendicular to it as it were. The form is a circle as this corresponds to U(1) symmetry (used in locally gauging the charged field, from which the EM arises when demanding that the Lagrangian of the charged Dirac field is invariant under this locally gauging; which was still unknown when Kaluza and Klein made their contribution, but the U(1), circle group, symmetry was already known in classical theory on EM, so; their whole framework is purely classical, but an analogue of it was used in string theory, the circle replaced by Calabi-Yau manifolds, to account for the symmetries of other interactions, and point particles replaced by strings) from which the field arises.
It was thought that particles, on top of their motion through large 3d space move on this these small circles too, to account for EM. I can't see though how a motion on these small circles results in a motion in 3d though. The EM force induces motion in 3d, and this isn't what is happening in the theory.
There is merely an extra dimension introduced to put information of the EM field on, and is not a real tiny spatial dimension. Of course you can say that when a particle moves through the bulk it moves in the small one too, and the 5d metric indeed gave rise to the Maxwell equations, but the motion in the curled up 5th dimension (,4th spatial), but the extra metric, being the components of the metric of the curled up circle stays just a mathematical one, containing the information of the field, by which the particle moves through the 3d bulk.
Likewise, the curled up dimensions of string theory are mathematical structures, like a Calabi-Yau manifold, which seemed to fit the bill to incorporate the U(1)×SU(2)×U(3) symmetries of the two other forces, of which the weak force (claimed to emerge from the breaking of the SU(1)×SU(2) symmetry by the Higgs field) might not be fundamental though.
All this extra stuff, together with strings and their vibrations, corresponding to charges, corresponds to a mathematical extension of space. Be it 6 or 26 dimension-like. The string offers a nice way out from that nasty concept of point particles.
More likely it seems that particles are curled up 3d structures, tiny Planckian spheres, obtained from curling three dimension of a 6d space up in a 7d space, like a circle on a cilinder. The can all be on top of each other without infinities occurring. The concept of a black hole singularity would disappear and only be approximate. An infinite self energy of charged particles would not have to be renirmalized, as it's not infinite.