  We realize the treatment of bound and continuum nuclear systems in the
proximity of a three-body breakup threshold within the ab initio framework of
the no-core shell model with continuum. Many-body eigenstates obtained from the
diagonalization of the Hamiltonian within the harmonic-oscillator expansion of
the no-core shell model are coupled with continuous microscopic three-cluster
states to correctly describe the nuclear wave function both in the interior and
asymptotic regions. We discuss the formalism in detail and give algebraic
expressions for the case of core+$n$+$n$ systems. Using
similarity-renormalization-group evolved nucleon-nucleon interactions, we
analyze the role of $^4$He+$n$+$n$ clustering and many-body correlations in the
ground and low-lying continuum states of the Borromean $^6$He nucleus, and
study the dependence of the energy spectrum on the resolution scale of the
interaction. We show that $^6$He small binding energy and extended radii
compatible with experiment can be obtained simultaneously, without recurring to
extrapolations. We also find that a significant portion of the ground-state
energy and the narrow width of the first $2^+$ resonance stem from many-body
correlations that can be interpreted as core-excitation effects.
