No, you cannot run Docker inside an OpenVZ container.
This is because OpenVZ uses a shared kernel architecture, and Docker requires functionality that is not available in OpenVZ’s kernel. In simple terms, OpenVZ doesn’t provide enough isolation to effectively run Docker containers inside of its own containers.
However, it is possible to run Docker on the host system that runs the OpenVZ containers. Alternatively, using a different virtualization technology that does support full kernel virtualization (like KVM, Xen, or even bare-metal Docker) might be an option.
