From a quick search on the copyright database, it seems the flag is from 1911, and generally states don't own IP and state flags (or country flags) aren't protected, and others have used the bear or the flag in new copyrighted works, so it seems that the bear image has fallen into the public domain and if it ever was protected by copyrighted, it isn't now.
As for its availability for use as a trademark, there are lots of bears in use, including grizzlies, so whether your bear representing your goods or services is confusingly similar to someone else's bear and their goods or services depends on what your intended use is and what's out there.
You need to consult an IP lawyer to fully disclose your plans and get some specifiic advice.
Disclaimer: Please note that this answer does not constitute legal advice, and should not be relied on, since each state has different laws, each situation is fact specific, and it is impossible to evaluate a legal problem without a comprehensive consultation and review of all the facts and documents at issue. This answer does not create an attorney-client relationship.
It's unlawful in California to offer business services by using an insignia, term, or symbol that could be construed to imply either a connection with a State agency or State approval or endorsement of those services. Bus. & Prof. Code § 17533.6(a).
In some instances, such conduct is a criminal offense. Penal Code §§472 and 538g(b) (unlawful to use state seal).
There are many images of bears out there. Using the particular bear image from California's state seal seems to me to imply that you're trying to associate the state with your business. That's unlawful.