Is SIREN number obligatory for all businesses in France?

Is it possible to be registered in France as a self-employed, and do not have a SIREN number? When someone claims that they are self-employed, but cannot present their SIREN number, can I safely assume that they are not really a business?

And a related question: if someone in France is not self-employed, can they provide services to businesses? Can they issue an invoice? What tax number would be on such invoice?