I'm asking this within the framework of brane theory. My intuition is that branes are objects with finite volume, but then how can they accomodate an Universe that is infinite in extent?
Branes typically have infinite volume. As an intuitive example of a brane, imagine an infinite plane in 3D, which is a 2-brane. The brane is specified by (for example), $z=0$ if $(x,y,z)$ are the usual Cartesian coordinates. An infinite line is a 1-brane and is specified by (again, for example) $y=z=0$. If the branes are Euclidean and wrapped up, then it is possible that they have finite volume.