The OSI model is just a model. That means it describes an idealized concept, rather than anything in the real world. So the question is really hypothetical. If you create a network with only one layer, you are not conforming to the OSI model, so at that point, the question becomes moot. Also, there's no strict definition of a network, local or otherwise.
The OSI model describes the component functions of a packet-switched network. Information is divided into parts called packets, and these are individually sent to the destination. The layers of the model are used to control and route the packets.
There are other kinds of networks that don't use packets (although they are rapidly becoming obsolete). Because there are no packets, there is no need for the functions of the OSI model (framing, routing, etc.)
One example of this is the (wired) telephone network**. It is (was) a circuit-switched network, where a temporary data path is set up between the endpoints. The data is sent in a continuous stream -- no packets. After all the data is sent, the data path is removed.
** In truth, I'm describing the telephone network of at least 30 years ago. Today, most telephone networks have converted to packet-switched networks because of their efficiency.