Fields in spacetime could be simultaneously discrete and continuous, in the same way that information can. It has been shown that the amplitudes phi(x(n)) that a field takes at a generic discrete set of points x(n) can be sufficient to reconstruct the field phi(x) for all x, namely, if there exists a certain type of natural ultraviolet (UV) cutoff in nature, and if the average spacing of the sample points is at the UV-cutoff scale. Here, we generalize this information-theoretic framework to spacetimes themselves. We show that samples taken at a generic discrete set of points of a Euclidean-signature spacetime can allow one to reconstruct the shape of that spacetime everywhere, down to the cutoff scale. The resulting methods could be useful in various approaches to quantum gravity.