Compartmentalization is a strategy of the cell to control matter distribution in space and time. Synthetic counterparts of natural compartments may provide a tool to emulate the structural organization and functional activity of cellular molecular machines. Here we present a semisynthetic DNA-protein nanomachine as a simplified but programmable model of the proteasome. This cell-free nanoscale model is composed of two connected DNA origami nanocages, one nanocage housing the segregase/unfoldase enzyme p97, and the other hosting the protease chymotrypsin. The unidirectional immobilization of p97 inside the DNA origami nanocages introduces a 'gateway' mechanism that governs the access to compartments and the direction in which substrates are processed. Our data indicate that the reaction rate of each individual enzyme increases upon its spatial confinement within the DNA nanocage. Moreover, their physical connection into a chimera further improves their performance, by coupling two catalytic processes and avoiding off-target proteolysis.  Due to its programmability and possibility to host various types of enzymes within the inner cavity, this nanomachine may enable to engineer and manipulate DNA-enzyme systems capable to perform complex reaction cascades, with promising applications in biomedical and biotechnological fields.