The Borromean He6 nucleus is an exotic system characterized by two halo neutrons orbiting around a compact He4 (or α) core, in which the binary subsystems are unbound. The simultaneous reproduction of its small binding energy and extended matter and point-proton radii has been a challenge for ab initio theoretical calculations based on traditional bound-state methods. Using soft nucleon-nucleon interactions based on chiral effective field theory potentials, we show that supplementing the model space with He4+n+n cluster degrees of freedom largely solves this issue. We analyze the role played by α clustering and many-body correlations, and study the dependence of the energy spectrum on the resolution scale of the interaction.