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You can define a Hausdorff Dimension on anything that has a (mathematically sound) metric by simply replacing the typical euclidean norm in its definition by the metric variant. So by finding some kind of distance measure - e.g. metric - that works on digits of numbers, no matter how arbitrary it is, you can just apply typical box-counting to arrive at an approximate Hausdorff Dimension - the box-counting dimension with respect to the chosen metric - of pi.

Finding the *actual* Hausdorff Dimension would be quite involved, I guess, possibly involving solving wether pi is normal, e.g. wether all digits occur equally often. That's a long standing open question.
Boxcounting could be done relatively quickly on the computer.