Both.
I’ve heard a theory that the entire universe is inside a black hole, which makes the “Finite, but unbounded,” understandable. In a black hole things fall forever, as space time is bent to a ever expanding infinite ‘space’, yet has a boundary.
The entire Universe being a black hole is equivalent to it being of spherical curvature and closed (with the singularity at the center being the “Big Crunch”). Until about 15 years ago, that was indeed a plausible hypothesis, but observations since then have pretty well ruled it out.