Generally when people ask this question they are talking about money, so lets dive right in.
After about an hour of research and number crunching, I've come up with the following data.
Federal and State spending in California is 640B, while Federal and State revenues are 570B. Of this, roughly 70B is the existing Federal deficit.
In short, California would be in the exact same position it is now; as part of the USA.
I've been rounding as some of the information I've found is inexact, but generally, the only "deficit" California would face is that currently existing on the Federal level.
There are a few ways to cut into this.
Tripling the state sales tax would cover the gap.
Doubling the state income tax would also do so, without the need to touch the federal income tax rates.
Upping all Federal income taxes by a third would also cover it.
As would cutting all military spending to 0.
Of course, each of those is silly on their own, and a combo would likely be used.
20B of the military spending is on spending outside the state; stuff like research and development for new projects California may not be interested in, or deployments overseas. That could easily be sliced in half.
Another roughly 25B could be raised by adding a 2.5% sales tax, bringing us halfway there.
California likely has little desire to remain as well armed as the US as a whole, as such the 50B military budget could easily be cut by a fifth.
It also makes little sense to have two overlapping income tax bracket systems, so abolishing one and adjusting the other seems prudent. Abolishing federal income taxes and multiplying the income tax rate for each state bracket by 3.7 would just about break even, meaning the current 1.00% tax bracket is now a 3.70% bracket, and the current 12.30% bracket is now a 45.51% bracket. However, by using a flat "4" and increasing the 1% bracket to 4%, and the 12.3% bracket to 49.2%, you can take another 20B off the deficit.
The remaining 5B likely could be found the way other small deficits are covered, small cuts to other programs. Remember that we are now talking about 5B out of 575B in spending.
So in short, yes. When it comes to the ability of the government to pay for services, California could indeed "work" as it's own country.