We construct well-balanced, high-order numerical schemes for one-dimensional blood flow in elastic vessels with varying mechanical properties. We adopt the ADER (Arbitrary high-order DERivatives) finite volume framework, which is based on three building blocks: a first-order monotone numerical flux, a non-linear spatial reconstruction operator and the solution of the Generalised (or high-order) Riemann Problem. Here, we first construct a well-balanced first-order numerical flux following the Generalised Hydrostatic Reconstruction technique. Then, a conventional non-linear spatial reconstruction operator and the local solver for the Generalised Riemann Problem are modified in order to preserve well-balanced properties. A carefully chosen suit of test problems is used to systematically assess the proposed schemes and to demonstrate that well-balanced properties are mandatory for obtaining correct numerical solutions for both steady and time-dependent problems.