Generally, a person claiming unemployment benefits (a “claimant”) is eligible for benefits if he or she is:
(1) out of work due to no fault of his or her own; AND
(2) physically able to work; AND
(3) actively seeking work; AND
(4) ready to accept work.
Details about eligibility for unemployment benefits can be found on the web site of the California Employment Development Department (EDD) here: http://www.edd.ca.gov/Unemployment/FAQ_-_Eligibility.htm#Whataretheeligibilityrequirements
In your situation, requirement #1 might be an issue. While you might have good cause to quit because you didn't have child care, precedent decisions hold that if the reason a claimant doesn't have child care is only due to the cost, then that is not good cause to quit. Also, EDD expects employees to try to get a leave of absence from work rather than to resign whenever a leave will resolve the problem, such as by allowing the employee to locate alternative child care.
See this page from the EDD web site:
http://www.edd.ca.gov/UIBDG/Voluntary_Quit_VQ_155.htm#Care of Children or Home
The Notice of Determination stating your claim was denied includes information about the appeal. You MUST file your appeal within 20 days of the date stated in that letter. Do not miss the deadline.
In the appeal, make a brief statement just one or two lines long saying why you believe the denial was incorrect. Save your detailed argument and evidence for the hearing. For example: "I was available to work full-time so should not be disqualified."
In a few weeks, you will receive a notice of an appeals hearing with the date, time and location. At the hearing, be prepared with as much evidence as possible. You should also know the law the administrative law judge will consider. You can get a lot of helpful information on the EDD website.
Summaries of the law (Benefit Determination Guide)
Precedent Decisions (law the administrative law judges rely on)
Frequently asked questions
Filing a claim for unemployment benefits
You can be represented by anyone at the hearing. If your appeal will be difficult or you are uncomfortable speaking, you may wish to retain an attorney to help you prepare or to represent you at the hearing. For training, expect the attorney to need approximately three hours. For representation at the hearing, expect the attorney to need three to seven hours to prepare, depending on the complexity of the case, witnesses, documents and other evidence, and allow two hours for the hearing itself. Unemployment hearings usually last one hour or less, but you must arrive early to look at the file, and there is a possibility you will have to wait past your hearing time if the previous case has not finished.
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I don't know California law specifically, so a CA lawyer would have to confirm this, but generally quitting to care for a family member is considered a voluntary quit for personal reasons and makes one ineligible for unemployment benefits.
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I'm not sure about CA, but North Carolina recognizes a "family hardship" exception that may allow parents to receive unemployment if they resign due to a lack of childcare alternatives. You should probably consult with a local attorney familiar with unemployment laws to see if this is an exception recognized in CA and if your facts would fit it. Good luck.