Caracals are legal in California provided you have a permit under 14 CCR § 671. Importation, Transportation and Possession of Live Restricted Animals. You’ll find a document on permits at § 671.1.
In summary, that is the answer to the question. You will have to apply for a permit for any wild cat species that you wish to possess and own in California except for the cheetah, which I find extraordinary. Yes, the cheetah is an exception to the general rule (important: check this as it is too odd to take on face value). Also, hybrids of domestic cats are not restricted. This implies that the first filial (F1 and a lower) wild cat hybrids are not restricted so you could own an F1 Savannah cat without a licence in California, for example.
I would recommend that a person wishing to own and possess a caracal or any other wild cat species in California contact their local authority and/or the California Department of Fish and Wildlife. Ironically, at this particular moment, while researching this answer, their website is down!
Below is a link to the Thomson Reuters Westlaw California Code of Regulations which lists 671 which states that “It shall be unlawful to….possess live animals restricted in subsection (c) below except under permit issued by the department”. Section 671.1 lists the purposes for which a permit shall be granted and which is subject to conditions and restrictions designated by the department.
As this is a complicated topic is better to speak to somebody on the phone because they will get to the point quickly, hopefully, rather than ploughing through a pile of technical documents which might baffle you.
Clearly, the law applies to individuals owning a caracal and zoos and breeders. And therefore there are different rules for different individuals and organisations. No doubt some individuals might like to own a caracal as a pet. My thought there is that only under exceptional circumstances is it a good idea. Although caracals are one of those wild cat species which are more predisposed than others to being a pet rather like the cheetah. Both the caracal and cheetah have a history of working with people and hunting with people. You can go back thousands of years and find records of both these wild cat species working with people as hunters.
I would cross-reference federal animal laws and more local animals laws i.e. city regulations, to make sure that there isn’t a clash. It is complicated I think. You have to be careful.