Unless expressly prohibited from owning firearms, then you are eligible.
TLDR, these are the requirements.
1) Must not be prohibited from purchasing a firearm
California DOJ has published a useful matrix for prohibiting categories of gun ownership. ATF also has its own list of requirements, but they are all covered by the CA DOJ's requirements.
Federal Eligibility Laws
Federal possession laws are defined in the U.S. federal code 18 U.S. Code Chapter 44 - FIREARMS.
The violation of these laws is punishable by up to ten years in prison, and the punishment gets considerably more severe if a repeat offender is committing them.
The federal rules can be found here, but here is the brief rundown:
Felons
Drug users or addicts
Aliens (meaning illegal aliens and aliens with non-resident visas)
Individuals subject to domestic restraining orders
Prior convictions of domestic assault
Fugitive from justice
Dishonorably discharged from the military
It is also completely illegal to purchase a firearm or ammunition for anyone who falls within one of these categories. You will get slapped with up to ten years in the prison, so don't even think about it. This is called a straw purchase, it is illegal, and if you get caught, you will do time. Don't buy a gun for anyone, ever. They can do it themselves, and if they can't it's because of something that will get you in trouble.
California Eligibility Laws
California has a tiered structure of firearms prohibitions, leading with lifetime prohibitions. For accuracy, the following are verbatim from the California firearms guide and penal code.
Lifetime Probitions
Any person convicted of any felony or any offense enumerated in Penal Code section 29905.
Any person convicted of an offense enumerated in Penal Code section 23515.
Any person with two or more convictions for violating Penal Code section 417, subdivision (a)(2) Any person adjudicated to be a mentally disordered sex offender. (Welf . & Inst . Code, § 8103, subd. (a)(1) .)
Any person found by a court to be mentally incompetent to stand trial or not guilty by reason of insanity of any crime, unless the court has made a finding of restoration of competence or sanity. (Welf. & Inst. Code, § 8103, subds. (b)(1), (c)(1), and (d)(1) .)
Ten Year Prohibitions
Any person convicted of a misdemeanor violation of the following: Penal Code sections 71, 76, 136 .5, 140, 148, subdivision (d), 171b, 171c, 171d, 186 .28, 240, 241, 242, 243, 244 .5, 245, 245 .5, 246, 246 .3, 247, 273 .5, 273 .6, 417, 417 .1, 417 .2, 417 .6, 422, 626 .9, 646 .9, 830 .95, subdivision (a), 17500, 17510, subdivision (a), 25300, 25800, 27510, 27590, subdivision (c), 30315, or 32625, and Welfare and Institutions Code sections 871 .5, 1001 .5, 8100, 8101, or 8103.
Five Year Prohibitions
Any person taken into custody as a danger to self or others, assessed and admitted to a mental health facility under Welfare and Institutions Code sections 5150, 5151, 5152; or certified under Welfare and Institutions Code sections 5250, 5260, 5270 .15. Persons certified under Welfare and Institutions Code sections 5250, 5260, or 5270 .15 may be subject to a lifetime prohibition pursuant to federal law.
There are additional prohibitions, which are available here. Please do yourself a service and read this in its entirety. We are summarizing it to take some of the legal speak out of it, but make sure to read it through on your own. Remember, you are the one who will be held responsible for your firearm.
2) Must be 21 years or older, unless specifically exempted (CA PEN 27510).
The Exemptions Include:
A person 18 years of age or older who possess a valid, unexpired hunting license issued by the Department of Fish and Wildlife.
An active peace officer, as described in Chapter 4.5 (commencing with Section 830) of Title 3 of Part 2, who is authorized to carry a firearm in the course and scope of his or her employment.
An active federal officer or law enforcement agent who is authorized to carry a firearm in the course and scope of his or her employment as a reserve peace officer.
A person who provides proper identification of his or her active membership in the United States Armed Forces, the National Guard, the Air National Guard, or active reserve components of the United States.
A Person who provides proper identification that he or she is an honorably discharged member of the United States Armed Forces, the National Guard, the Air National Guard, or active reserve components of the United States.
The exemptions apply to the sale, supplying, delivery, transfer, or giving possession or control of a firearm that is not a handgun to a person 18 years of age or older.
18-20-year-olds are currently not able to purchase semi-automatic long guns. This is now being challenged through the courts.
3) Must have a valid Firearms Safety Certificate (FSC)
On January 1, 2015, the state of California started to require firearms purchases to obtain and provide to the dealer an FSC before the transfer of a firearm.
To obtain the FSC, you must pass a written test on firearm safety and laws specified by CA State Law. At Dumpling Defense, we can administer this test to customers purchasing a firearm from us or using any of our services.
For reference, this is CA DOJ-provided study guide. The FSC needs to be renewed every 5 years.
There are cases where the purchaser can be exempt from needing an FSC.
4) Must provide ID, proof of residency, and supplemental documentation if needed
California identification (driver’s licenses) are issued in two types: “REAL ID” and “Federal Limits Apply.” If the individual's ID is labeled as Federal Limits Apply, firearms dealers should require them to provide proof of lawful presence.
The address on the valid ID must be current, but if it is not current, the buyer must bring proof of address, such as a utility bill or lease agreement.
5) Must pass CA DOJ DROS background check
During the sale of a firearm, CA DOJ requires Dumpling Defense to fill out a Dealer Record of Sale (DROS). The DROS information is entered digitally and will run through CA DOJ's background check to verify the purchaser's residency in California and the purchase's eligibility.
This step is done at the time of purchasing, followed by a CA DOJ-required 10-day holding period where CA DOJ will run the background check and respond to us with the purchase eligibility of the purchaser. The countdown starts at the time of DROS submission.
If CA DOJ returns no red flags, the purchaser can pick up the firearm after the 10-day holding period.
There are rare cases where the firearm can be held longer. There are also cases where the 10-day waiting period can be exempted.