In Riemannian geometry (i.e. GR) each point has some local invariants associated to it, a famous one being curvature. "Local" here means that it only depends on the metric in an arbitrarily small disc/ball around the point. So the answer is no, in GR you can't always change coordinates so that space is locally flat. (This is in contrast to symplectic geometry for example, where each point does look locally like the model space).