In this article we study a \emph{non-directed polymer model} on Z, that is a one-dimensional simple random walk placed in a random environment. More precisely, the law of the random walk is modified by the exponential of the sum of ``rewards'' (or penalities) βωx−h sitting on the range of the random walk, where (ωx)x∈Z are i.i.d.\ random variables (the disorder), and where β≥0 (disorder strength) and h∈R (external field) are two parameters. When β=0,h>0, this corresponds to a random walk penalized by its range; when β>0,h=0, this corresponds to the ``standard'' polymer model in random environment, except that it is non-directed. In this work, we allow the parameters β,h to vary according to the length of the random walk, and we study in detail the competition between the \emph{stretching effect} of the disorder, the \emph{folding effect} of the external field (if h≥0), and the \emph{entropy cost} of atypical trajectories. We prove a complete description of the (rich) phase diagram. For instance, in the case β>0,h=0 of the non-directed polymer, if \go_x ha a finite second moment, we find a transversal fluctuation exponent ξ=2/3, and we identify the limiting distribution of the rescaled log-partition function