Does the math in string theory tell us anything about the size of any of the extra dimensions?
no so much.
for one thing there are many models with varying 'extra dimensions'....sometimes
Calabi-Yau [which I think are small, 'compactified'] and other times branes [bigger] and sometimes a mix....Then there is an ADD model with large extra dimensions.
Some discussion on extra dimensions here:
edit: a discussion with some background...no precise answer...here:
This discussion brings out some of the complexities in addressing your question
To a first approximation, string theory in its current form tells us nothing. Instead, you tell it what you want.
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