Yes, S' is infinite. On the other hand, all numbers in S' are rational, so it is countable. So, you are right, the cardinality of S' is .
Let S be the set of all real numbers in the interval (0, 1) whose decimal expansions contain only 0's, 2's and 7's. Let S' be the elements of S whose decimal expansions contain only finitely many 2's and 7's. What is the cardinality of S'?
I think it is because even if it contains finitely 2's and 7's you could still make infinity numbers, or am I wrong? For example you could always tack on another 2 or 7 to make a new number, which still has a finite number of digits.