It depends on what exactly you have in mind. dfx start only spins up a single replica, which can be either in NNS (aka system) mode or in normal application subnet mode.
This is not super trivial to do, but definitively achievable. You can look at this file and the other scripts in that folder to see how our NNS frontend team spins up their local NNS instance for testing the frontend.
If you only want certain canisters of the NNS, have a look here for the individual NNS canisters. They all should have instructions in their README how to deploy them locally.