Is the universe infinite? Or is it finite?
You can relate observational measurements to this question by making an assumption. Namely, if we assume that the cosmological principle holds, then the curvature of the universe that we have measured from our earthly vantage point is true throughout the universe. This allows to extrapolate our measurement of the local curvature so as to make predictions about the global shape of the universe. If the curvature is $\Omega > 1$, then the universe's spatial shape can be thought of as the equivalent of the surface of a sphere in 4-D Minkowski space. In this case it is finite. If we have $\Omega < 1$ then it has a hyperbolic shape, making it infinite. If $\Omega = 1$ then it is flat, also making it infinite. I won't go into the subtleties of discussing the shape of the universe in cases where curvature is not constant or not necessarily the same in all directions, suffice it to say that things can get more intricate than this simple 3-way scenario. Current observations don't actually allow us to distinguish between these 3 scenarios as we currently roughly measure $\Omega = 1 \pm 0.01$. This value that is very close to flatness is speculated to be due to inflation, but that is another topic for another time.
But if it's finite, the idea that we can't go beyond a certain space just creeps me out.
This is just plain wrong. If the universe is finite that doesn't mean that there exists a clear cut "boundary" to it. Indeed, that would be quite odd as you rightfully point out. Using the example of a finite space given above, you can see how a universe can be finite without running into this problem. Imagine the surface of a sphere (I stress surface and not the space inside it), like say the surface of the globe. It is a finite space, in that it has a finite surface, that doesn't however require it to have a limit point where you would "exit" the space were you to go beyond it. You can easily generalize this to higher dimensions. Note that the extra dimension used here (we imagined a 2-D surface by invoking the surface of a 3-D object) is merely a pedagogical and doesn't actually have to exist. Imagine instead a 2-D world map to realize that there is no obligation to invoke an extra dimension to describe this space, it just makes it easier to visualize.
Hope that answers your question and you're not as creeped out anymore.