Bobcats are NOT legal pets in California unless you have a ‘special permit’. All cats species other than the domestic cat are classified as “not normally domesticated in this state” and for that reason the state has decided that “it shall be unlawful to import, transport, or possess live animals restricted in subsection (c). Subsection (c) is a long list of wild animals including the family of cats – all species of cat (scientific term: Felidae). But domestic cats and hybrids of domestic cats are not restricted (makes sense seeing as they are domesticated!).
The state considers the cheetah as being a threat to native wildlife and the agricultural interests of the state or to public health and safety and is termed a ‘detrimental animal’. You require a permit to possess one for that reason.
I suspect that getting a permit is going to be tricky and you’re going to have to demonstrate that you have sufficient funding, know-how, facilities, time and commitment et cetera to possess any wild cat species including the bobcat as a pet. And you will probably have to renew the permit annually and pay a fee but please check the relevant authorities which is probably California Department of Fish and Wildlife.
The relevant law is: § 671. Importation, Transportation and Possession of Live Restricted Animals.
If you search in Google for that or simply “14 CCR § 671” you will find the law in full if you want to look at any other animals that are illegal to own without a permit.
Below are some articles on animal laws.