We consider a one-dimensional, time-reversal-invariant system with intrinsic attractive interactions. Such a system is gapless due to the strong quantum fluctuations of the superconducting order parameter. However, we show that a sharply defined topological phase with protected, exponentially localized edge states still exists. If one of the spin components is conserved, the protection of the edge modes can be understood as a consequence of the presence of a spin gap in the bulk. In the more general case, the localization of the edge states arises from a gap to single particle excitations in the bulk. We consider specific microscopic models for the quantum wire and demonstrate both analytically and numerically (using density matrix renormalization group calculations) that they can support the topologically non-trivial phase.