The answer is ‘no’, as shown by many people here.
There are two ways of trying to make sense of the ‘set of all sets’, although not in the literal sense. If you are looking for a model of ZFC, then you can consider the collection of all sets whose rank is less than some inaccessible cardinal (assuming that one exists at all). This collection is, of course, still a set in the universe, and it yields a standard model of ZFC. However, inaccessible cardinals are uncountable, so this approach does not yield a countable ‘set of all sets’. Note the quotation marks!
However, if ZFC is consistent, then there exists a countable model of ZFC, which is a consequence of the Löwenheim-Skolem Theorem from logic and model theory. Therefore, you may want to take this model to be your ‘set of all sets’.
Whatever it is, from the model-theoretic point of view, this ‘set of all sets’ is never going to be an element of itself.