We show that multiple Weyl and double-Weyl points (DWPs) arise in a chiral elastic system through stacking two-dimensional honeycomb mechanical structures. On the distinct kz plane, the band structures calculated from a tight-binding (TB) model exhibit Weyl points (WPs) at Brillouin vertices and DWPs at Brillouin centres. Based on the TB model, a practical chiral mechanical structure can be constructed by 3D printing technology. The numerical calculation illustrates several Weyl and DWPs as expected in our analysis of the TB model. To verify the topological feature, the topological charge of degeneracy is calculated. Within these WPs, we theoretically prove the existence of topologically protected surface modes, and their robustness against defects is also demonstrated.