Not necessarily. You could theoretically write out a transaction on a piece of paper. You could calculate the transaction hash and the signatures by hand. Then the transactions can be sent by runners to other people who copy all of the data onto more paper and give them to more runners to give to more people and so on and so forth. If we had wires then there wouldn't need to be runners, it could be done over morse code or something similar.

In this scenario a node would be a person and the network be all of the runners going between nodes.

Of course this is entirely unfeasible as each person would have to calculate the hash of a transaction and verify the transaction by hand whenever it receives a new one and that would take ages to do.