Studies of a class of infinite 1D self-gravitating systems have highlighted that, on one hand, the spatial clustering which develops may have scale-invariant (fractal) properties and, on the other hand, they display ‘self-similar’ properties in their temporal evolution. The relevance of these results to 3D cosmological simulations has remained unclear. We show here that the measured exponents characterizing the scale-invariant non-linear clustering are in excellent agreement with those derived from an appropriately generalized ‘stable-clustering’ hypothesis. Further an analysis in terms of ‘haloes’ selected with a friend-of-friend algorithm reveals that such structures are, statistically, virialized across the range of scales corresponding to scale invariance. Thus the strongly non-linear clustering in these models is accurately described as a virialized fractal structure, very much in line with the ‘clustering hierarchy’ which Peebles originally envisaged qualitatively as associated with stable clustering. If transposed to 3Ds these results would imply, notably, that cold dark matter haloes (or even subhaloes) are (1) not well modelled as smooth objects and (2) that the supposed ‘universality’ of their profiles is, like apparent smoothness, an artefact of poor numerical resolution.