Nanometre-scale pores and capillaries have long been studied because of their importance in many natural phenomena and their use in numerous applications1. A more recent development is the ability to fabricate artificial capillaries with nanometre dimensions, which has enabled new research on molecular transport and led to the emergence of nanofluidics2,3,4. But surface roughness in particular makes it challenging to produce capillaries with precisely controlled dimensions at this spatial scale. Here we report the fabrication of narrow and smooth capillaries through van der Waals assembly5, with atomically flat sheets at the top and bottom separated by spacers made of two-dimensional crystals6with a precisely controlled number of layers. We use graphene and its multilayers as archetypal two-dimensional materials to demonstrate this technology, which produces structures that can be viewed as if individual atomic planes had been removed from a bulk crystal to leave behind flat voids of a height chosen with atomic-scale precision. Water transport through the channels, ranging in height from one to several dozen atomic planes, is characterized by unexpectedly fast flow (up to 1 metre per second) that we attribute to high capillary pressures (about 1,000 bar) and large slip lengths. For channels that accommodate only a few layers of water, the flow exhibits a marked enhancement that we associate with an increased structural order in nanoconfined water. Our work opens up an avenue to making capillaries and cavities with sizes tunable to ångström precision, and with permeation properties further controlled through a wide choice of atomically flat materials available for channel walls.