It’s ill-defined in the sense that it doesn’t uniquely define the set. There are at least two different sets that D could be (one containing it and one not containing it), hence the expression doesn’t denote a well-defined set. (*)

The axioms of ZF do not allow to form that expression, so the set doesn’t exist in ZF.

(*) This is from a universist view. In a pluralist view, one wouldn’t say that the fact of the matter of whether D contains itself or not is independent from naive set theory, and that there are set universes where it is the case and others where it isn’t. But I would hold that naive set theory starts from a universist view.

... and, as such, it doesn't contain itself!