I can't speak specifically to CA, but in all other states that I am aware of, you can start practicing without supervision as soon as you are sworn in. Countless law grads do that as a solo.
So, I am a corporate lawyer who represents startups. I didn't go work for a startup when I graduated; I practiced at large law firms for a very long time before that. Let me give you some advice:
1. Are you going to be a W-2 employee or a 1099 contractor of some sort? I ask because you said "startups" (plural). Are they affiliated with each other, like a holding company and operating subsidiary? If you would be a contractor, you are basically a solo practitioner with those startup clients, and would really need malpractice insurance (especially if the startups have outside investors).
You'd have some serious exposure. Even if you ultimately prevail in shareholder lawsuits, your legal fees would be astronomical. Thus, the need for malpractice. However, for someone with zero experience, I suspect that your premiums would be expensive, though not totally unaffordable. The problem is that your startup comp would be low so paying the premiums might be very painful.
2. One of the best benefits of starting at BigLaw (and to a lesser extent, a medium or small firm) is world-class training. You have several really smart and experienced people looking over your shoulder to make sure you are not f#$king up. You can learn from partners' and senior associates' comments, and by discussing the comments with them. You get to observe (and ultimately negotiate smaller stuff on your own until you are "running a deal") how to negotiate in corporate deals.
If you start under the circumstances you describe, you are gonna have to wing everything. You might say no problemo I'll just Google, but you know the saying, "You don't know what you don't know." If you don't have adequate experience and knowledge base, you cannot spot issues and that very often leads to massive f&*k ups. You'd have no outside counsel to watch your back or to ask questions (presumably).
In this regard, let me advise you about a Section 83(b) election under the Internal Revenue Code. It basically allows founders and others to pay taxes upfront on equity (restricted stock and founder stock subject to vesting) . . . . as in, you pay taxes based on ,say, $0.05 per share valuation as opposed to $10.00 valuation later on ( x number of shares). It could cost founders and other grantees literally millions and millions in additional taxes. And you only have 30 days to make the election! A total disaster if you fail to advise the founders about this on a timely basis. The point is, if you don't have experience, you could easily miss something so basic and devastating like this. Just one example. Don't you ever forget about 83(b) election (it's not usually applicable to stock options btw).
3. Having said all that, there are a lot of upsides to what you are contemplating. The cash comp would be low, I am sure, but if you get equity (insist on it), it could be worth beaucoup bucks down the road. Like millions, potentially. Beware of tax issues with vesting. Of course, most startups self-implode so be realistic about your chances.
Instead of telling your clients (company/founders) that they can't do something, you get to tell them how to craft a solution to solve a knotty problem. You get to know your company intimately in every respect. They might include you in every important business decision. You get to be a quasi-lawyer, quasi-businessman/woman. . . . a true trusted advisor. It *could* be totally awesome in the right situation.
In conclusion, I would say that ideally you would want to have practiced at least 2+ years at a real firm before going in-house like this (unless you have experienced attorneys above you to oversee your stuff).
Good luck!