Yes, even without a signed document, if you and the employer agree to work together and there is an agreement to pay you for your services, it constitutes a contract of employment.
As long as you perform your work or make yourself available and willing to work, you are entitled to legal protection.
According to the Basic Conditions of Employment Act (BCEA) and Labour Relations Act (LRA), the employer must provide written particulars of the employment when you start working.