We present a mechanical model of tissue homeostasis that is specialised to the intestinal crypt. Growth and deformation of the crypt, idealised as a line of cells on a substrate, are modelled using morphoelastic rod theory. Alternating between Lagrangian and Eulerian mechanical descriptions enables us precisely to characterise the dynamic nature of tissue homeostasis, whereby the proliferative structure and morphology are static in the Eulerian frame, but there is active migration of Lagrangian material points out of the crypt. Assuming mechanochemical growth, we identify the necessary conditions for homeostasis, reducing the full, time-dependent system to a static boundary value problem characterising a spatially heterogeneous "treadmilling" state. We extract essential features of crypt homeostasis, such as the morphology, the proliferative structure, the migration velocity, and the sloughing rate. We also derive closed-form solutions for growth and sloughing dynamics in homeostasis, and show that mechanochemical growth is sufficient to generate the observed proliferative structure of the crypt. Key to this is the concept of threshold-dependent mechanical feedback, that regulates an established Wnt signal for biochemical growth. Numerical solutions demonstrate the importance of crypt morphology on homeostatic growth, migration, and sloughing, and highlight the value of this framework as a foundation for studying the role of mechanics in homeostasis.