We construct and study a family random continuum polymer measures $\mathbf{M}_{r}$ corresponding to limiting partition function laws recently derived in a weak-coupling regime of polymer models on hierarchical graphs with marginally relevant disorder. The continuum polymers, which we refer to as directed paths, are identified with isometric embeddings of the unit interval $[0,1]$ into a compact diamond fractal with Hausdorff dimension two, and there is a natural 'uniform' probability measure, $μ$, over the space of directed paths, $Γ$. Realizations of the random path measures $\mathbf{M}_{r}$ exhibit strong localization properties in comparison to their subcritical counterparts when the diamond fractal has dimension less than two. Whereas two paths $p,q\in Γ$ sampled independently using the pure measure $μ$ have only finitely many intersections with probability one, a realization of the disordered product measure $ \mathbf{M}_{r}\times \mathbf{M}_{r}$ a.s. assigns positive weight to the set of pairs of paths $(p,q)$ whose intersection sets are uncountable but of Hausdorff dimension zero. We give a more refined characterization of the size of these dimension-zero sets using generalized (logarithmic) Hausdorff measures. The law of the random measure $\mathbf{M}_{r}$ cannot be constructed as a subcritical Gaussian multiplicative chaos because the coupling strength to the Gaussian field would, in a formal sense, have to be infinite.