We propose a novel one-dimensional simple model without disorder exhibiting slow dynamics and aging at the zero temperature limit. This slow dynamics is due to entropic barriers. We exactly solve the statics of the model. We derive an evolution equation for the slow modes of the dynamics which are responsible for the aging. This equation is equivalent to a random walker on the energetic landscape. This latter elementary model can be solved analytically up to some basic approximations and is shown to present aging by itself, as well as a slow logarithmic relaxation of the energy: ⪉ngle erangle(t) sim 1/ln(t) at large t.