We show that the elasticity Hilbert complex with mixed boundary conditions on bounded strong Lipschitz domains is closed and compact. The crucial results are compact embeddings which follow by abstract arguments using functional analysis together with particular regular decompositions. Higher Sobolev order results are also proved. This paper extends recent results on the de Rham Hilbert complex with mixed boundary conditions from Pauly and Schomburg (2021, 2022) and recent results on the elasticity Hilbert complex with empty or full boundary conditions from Pauly and Zulehner (2020, 2022).