Don't confuse responsibility with having an absolute answer. Self-possession and self-responsibility are not the same. There is no unified self to grasp, but responsibility need not mean Cartesian self-transparency or Platonic founding. There can be responsibility, being called to answer, without a demand for totalizing foundations. And responsibility is needed where we are today. It is somewhat (only somewhat!) analogous to the responsibility we bear for the causal consequences of our actions. These are also impossible to totalize or to assume. But for all that we still look where going when we are driving and are careful when turning on switches. So too we should be responsible in the sense of being aware of concepts and categories, and showing that awareness. This is not necessarily answering from some fixed ground, but it is knowing something about where we seem to be.
(I tried, with mixed success, to talk about issues of our finitude and responsibility in my book Postmodern Sophistications (Chicago: University of Chicago Press, 1990), which deals with history and tradition in the contexts of philosophy and of architecture.)