Yes, thankfully, we are, although limitations and restrictions usually alter from municipality to municipality. For example, at the time of writing, you may have up to ten hens in Johannesburg and only five in Cape Town. You may not keep roosters.
Please be aware that rules might change from time to time. Please call your local health inspector or read the bylaws online for absolute clarity.
Cape Town: Policies, bylaws and publications
Durban: eThekwini Open By-laws
Bloemfontein: Municipal By-Laws
Pretoria: By-Laws City of Tshwane
Port Elizabeth: Nelson Mandela Bay Municipality (By-laws relating to keeping of animals)