Engineering effective p-wave superconductors hosting Majorana quasiparticles (MQPs) is nowadays of particular interest, also in view of the possible utilization of MQPs in fault-tolerant topological quantum computation. In quasi-one-dimensional systems, the parameter space for topological superconductivity is significantly reduced by the coupling between transverse modes. Together with the requirement of achieving the topological phase under experimentally feasible conditions, this strongly restricts in practice the choice of systems which can host MQPs. Here, we demonstrate that semiconducting carbon nanotubes (CNTs) in proximity with ultrathin s-wave superconductors, e.g., exfoliated NbSe2, satisfy these needs. By precise numerical tight-binding calculations in the real space, we show the emergence of localized zero-energy states at the CNT ends above a critical value of the applied magnetic field, of which we show the spatial evolution. Knowing the microscopic wave functions, we unequivocally demonstrate the Majorana nature of the localized states. An effective four-band model in the k-space, with parameters determined from the numerical spectrum, is used to calculate the topological phase diagram and its phase boundaries in analytic form. Finally, the impact of symmetry breaking contributions, like disorder and an axial component of the magnetic field, is investigated.