The literal shape of space became a topic with Einstein's theory of general relativity. Before that, there had certainly been arguments about whether space was unbounded or had some limit, but they were mostly philosophical. In general relativity space-time is a mathematical manifold whose geometry is linked to observable features such as gravity, the expansion of the universe - and also lightcones, event horizons and singularities. This meant that (1) the theory now included various forms of boundaries set by the geometry, and (2) that the theory clearly leaves certain things "outside".
The first category of boundary are benign in the sense that they denote either regions that cannot influence each other but have the same kind of physics (we can never observe beyond a certain distance, we can never see the interior of a black hole from the outside), or represent parts of physics where the theory clearly breaks down (and "knows" it) like gravitational singularities or the big bang singularity. How to resolve those breakdowns is of course a rich frontier for theoretical physics, astrophysics and cosmology.
The second category is subtler. Einsteinian gravity is strongly based on the profound insight in Gauss' Theorema Egregium that the geometry of a manifold can be described only using properties found in the manifold, with no recourse to embedding it into a bigger outside space. The perennial question "what is outside the universe" is simply removed: general relativity does not need an outside, doesn't talk about an outside, and doesn't benefit from having an outside. That doesn't mean one cannot construct theories where space-time manifolds are embedded in other spaces but they are theories that need to be motivated rather strongly - there is no obvious evidence making them necessary, so Ockham's razor wants to shave them off.
Similarly quantum mechanics also adds boundaries but in the information domain: certain things are measurable only in a partial sense, and while the part that cannot be measured can still be described nicely mathematically the theory shows that we cannot know its state (in a pretty specific way).
To sum up, there are some things we know we cannot know or interact with but are not particularly worrisome: in a large accelerating expanding universe there are visible galaxies we can never reach, and there are galaxies that will forever be beyond our horizon. We however expect astrophysics to work the same over there. There are also limits to knowledge that are more annoying: the interior of black holes, what is beyond the CMB radiation. Then there are the theoretical boundaries where quantum gravity or other new insights we do not have yet are needed to continue inquiry. And finally there are the "boundaries" where we simply have no information, like "outside the universe" or some multiverse theories. Different people work on different things - philosophers and theoretical physicists are having fun in the last category (and arguing about how one can rigorously argue about such things), while most astrophysicists prefer to work in the data-rich regions and use it to extrapolate that is likely true for the other regions.
(James K. cited Wittgenstein's adage about being silent, but it is worth noting that this is usually interpreted more as a challenge than advice by many theoreticians.)