Quantum dots defined in carbon nanotubes are a platform for both basic scientific studies(1-5) and research into new device applications(6). In particular, they have unique properties that make them attractive for studying the coherent properties of single-electron spins(7-11). To perform such experiments it is necessary to confine a single electron in a quantum dot with highly tunable barriers(1), but disorder has prevented tunable nanotube-based quantum-dot devices from reaching the single-electron regime(2-5). Here, we use local gate voltages applied to an ultraclean suspended nanotube to confine a single electron in both a single quantum dot and, for the first time, in a tunable double quantum dot. This tunability is limited by a novel type of tunnelling that is analogous to the tunnelling in the Klein paradox of relativistic quantum mechanics.