In the paper, we investigate statistical and topological properties of fractional Brownian polymer chains, equipped with short-range volume interactions. The attention is paid to statistical properties of collapsed conformations with the fractal dimension D-f >= 2 in the three-dimensional space, which are analyzed both numerically and via the mean-field Flory approach. Our study is motivated by an attempt to mimic the conformational statistics of collapsed unknotted polymer rings, which are known to equilibrate into the compact hierarchical crumpled globules (CG) with D-f = 3 at large scales. Replacing the topologically stabilized CG state by a self-avoiding fractal path adjusted to the fractal dimension D-f approximate to 3, we tremendously simplify the problem of generating the CG-like conformations since we wash out the topological constraints from the consideration. Using a combination of the Flory arguments and Monte Carlo simulations of the chains with the effective fractional Brownian motion (fBm) Hamiltonian and volume interactions, we derive the dependence of the critical exponent on the fractal dimension of the non-perturbed fBm chain. We show that, with the increase in D-f, typical conformations become more territorial and less knotted. Distributions of the knot complexity, P(chi), for globular ring chains with D-f >= 2 suggest a direct correspondence between the fractal dimension and knotting of fractal paths with the finite excluded volume.