I think we can construct a Cantor set A where both A and 1/A are comprised entirely of 0-free numbers, which suffices. (I haven't proof checked this approach)

The idea is as follows :

Let 0 < a_0 < a_1 have only 2s, 5s, and 8s as digits and [1/a_1, 1/a_0] contains uncountably many numbers with only 2s, 5s, and 8s as digits, as well as uncountably many with only 2s and 8s.

Now given any interval in the above form we can always find a_01 , a_10 , with only 2s, 5s, and 8s, with a_0 < a_01 < a_10 < a_1 and such that every number in (1/a_10 , 1/a_01 ) has at least one 5.

We can then follow the standard procedure from there to construct a cantor set from intersecting nested closed sets, and observe that what is left must have the properties we need.