First, don't misunderstand: the Big Bang Theory, as a theory about the evolution of the Universe, has thousands of confirming observations.
The Big Bang "singularity" however, is an artifact of the Theory of General Relativity. Extrapolating the expansion of the Universe backwards, in the equations of General Relativity, leads to an infinitely dense, infinitely hot singularity a finite time in the past (13.7 billion years ago). It seems likely, however, that those "infinities" cannot actually exist in reality. Quantum gravity processes (that we know must exist but don't understand) will intervene and prevent an actual singularity.
We do not know if the Big Bang occurred in the context of an external multi-dimensional manifold that encloses our Universe. The Big Bang might have been an event that occurred inside some larger space --- we don't know. If so, that external space has been pushed aside by the expansion of "our" Big Bang, and is no longer accessible to us.