Whilst the Vela pulsar and its associated nebula are often considered as the archetype of a system powered by a ~10^4 year old isolated neutron star, many features of the spectral energy distribution of this pulsar wind nebula are both puzzling and unusual. Here we develop a model that for the first time relates the main structures in the system, the extended radio nebula (ERN) and the Xray cocoon through continuous injection of particles with a fixed spectral shape. We argue that diffusive escape of particles from the ERN can explain the steep Fermi-LAT spectrum. In this scenario Vela X should produce a distinct feature in the locally-measured cosmic ray electron spectrum at very high energies. This prediction can be tested in the future using the Cherenkov Telescope Array (CTA). If particles are indeed released early in the evolution of PWNe and can avoid severe adiabatic losses, PWN provide a natural explanation for the rising positron fraction in the local CR spectrum. Show Partial Abstract