No, as an individual citizen of California it is illegal to own a purebred wolf (100% wolf DNA). I’m presuming that the question is being asked by an individual person who wants to keep a wolf as a pet.
The law is different if you want to own a wolf-dog hybrid (a wolfdog). There are different levels of wolf-dog hybrid. A first filial hybrid is one in which the father is a wolf and the mother is a domestic dog or vice versa. This is an F1 hybrid. The dog is 50% wolf. If your dog is 50% wolf you will need a permit from the Department of Fish and Game in California. They only issue permits to qualified individuals and institutions such as research institutions. I would advise you to telephone the department and ask how they interpret the phrase “qualified individual”.
You won’t need a permit if your wolf-dog hybrid is a second-generation animal or lower. By lower I mean third, fourth, fifth generation and so on. Each generation further down the line has less wolf DNA in the animal. Second-generation wolf-dog hybrid is an F2 or second filial.
It has occurred to me that a person asking the question in the title may not know the amount of wolf DNA in the dog that they wish to adopt as a pet. This is because there is informal mating between wolves and dogs producing hybrids. This happens more commonly near human settlements where the population density of wolves is low and where dogs are common. I wonder whether a DNA test could be arranged. There would have to be some evidence that the animal was part wolf and part domestic dog.
That, as I understand it is the full picture. I have used the language of hybrid cats to describe the various filial levels.
FYI – the first recorded deliberate breeding of a wolfdog occurred in the Britain in 1766 when a male wolf mated with a dog which appears to have been a Pomeranian.