  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, $\mu$, over the space of directed
paths, $\Gamma$. 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 $p,q\in \Gamma$ sampled independently using the pure measure $\mu$ 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.
