There is much interest in developing synthetic analogues of biological membrane channels with high efficiency and exquisite selectivity for transporting ions and molecules. Bottom-up and top-down methods can produce nanopores of a size comparable to that of endogenous protein channels, but replicating their affinity and transport properties remains challenging. In principle, carbon nanotubes (CNTs) should be an ideal membrane channel platform: they exhibit excellent transport properties and their narrow hydrophobic inner pores mimic structural motifs typical of biological channels. Moreover, simulations predict that CNTs with a length comparable to the thickness of a lipid bilayer membrane can self-insert into the membrane. Functionalized CNTs have indeed been found to penetrate lipid membranes and cell walls, and short tubes have been forced into membranes to create sensors, yet membrane transport applications of short CNTs remain underexplored. Here we show that short CNTs spontaneously insert into lipid bilayers and live cell membranes to form channels that exhibit a unitary conductance of 70-100 picosiemens under physiological conditions. Despite their structural simplicity, these 'CNT porins' transport water, protons, small ions and DNA, stochastically switch between metastable conductance substates, and display characteristic macromolecule-induced ionic current blockades. We also show that local channel and membrane charges can control the conductance and ion selectivity of the CNT porins, thereby establishing these nanopores as a promising biomimetic platform for developing cell interfaces, studying transport in biological channels, and creating stochastic sensors.