(Expanding on Yuan's point about the irrelevance of "machine language":language" and JT's about "the wrong question to ask":)
In a sense, one oughtn't even be able to ask whether two topological spaces (i.e., objects of Top) are equal, only whether they are isomorphic; if one can't ask about equality, then one certainly can't speak of cardinality (with respect to such equality), and so there is no notion of the cardinality of the collection of objects isomorphic to a given one.
However, if you construe every topological space as implicitly carrying extra non-topological information via which such a non-topological notion of equality is defined (e.g., if you take the points of topological spaces to furthermore be elements of the cumulative hierarchy of well-founded sets of sets of sets..., allowing one to ask whether points in distinct spaces are equal by appeal to this extra structure, and accordingly whether spaces themselves are equal by virtue of an isomorphism sending points to equal points), then, of course, the question can be answered (in the given example, as noted above, the answer will be that the isomorphism classes form proper classes). But this is not really a question about the category of topological spaces, as such; this is a question about the particular manner in which one may choose to realize the intuitive theory of topological spaces within the ontology of another (meta)theory/implement the structure of the category of topological spaces within a context imposing further structure as well.
If one avoids selecting such "implementation details", then the question is meaningless, in precisely the same way as questions such as "Is the integer 9 an element of the rational -3/5?" are meaningless in the abstract.