A manifold for Hilbert's hotel

If $x_n$ are such points then $R=d(x_1,x_n) >0$ for all $n$ Hence $x_n\in B_R(x_1)$ Since $B_R(x_1)$ is compact so there is a convergent subsequence. Hence it does not happen.

For posterity here's an implementation of Micah's suggestion.

Let $(\rho, \theta, \phi)$ denote spherical coordinates on the open unit ball, $\Omega = d\phi^{2} + \sin^{2}\phi\, d\theta^{2}$ the round metric on the unit sphere, and $f(\rho) = \rho/(1 - \rho)$ (or any smooth, monotone function defined for $0 \leq \rho < 1$ with $f(0) = 0$ and, as $\rho \to 1^{-}$, $f \to \infty$ rapidly enough that $f$ is not improperly integrable).

In the metric $$ g = d\rho^{2} + f(\rho)^{2}\, \Omega, $$ the distance from the origin to the boundary of the ball is unity, but the volume element is $$ dV = f(\rho) \sin\phi\, d\rho\, d\theta\, d\phi. $$ Geometrically, the intrinsic radii of spherical shells centered at the origin grow rapidly enough that the volume is infinite.

In this universe, there exist countably many "cells" of fixed volume (though not suitable as three-dimensional hotel rooms, as asymptotically they necessarily become "intrinsically thin in the radial direction"), but any two rooms (or points) are separated by a distance of at most $2$ because the origin is at most one unit away from each room.