The rule isn't just California, it is pretty much everywhere. Individual wires have insulation (except ground or in certain cases neutral) that protects against zapping people/short circuits, but they need additional protection from all sorts of damage, though how much extra protection they need varies by situation.
The two choices are cable and conduit. There are a number of different types of cable, with the biggest difference being that some are usable in dry locations (therefore mostly indoors) and some can be used in wet locations. Cables all basically look the same, and you can't turn individual wires into cable.
Conduit includes two types of PVC - schedule 40 and schedule 80, as well as a few types of metal conduit and a few other possibilities.
Your options here are to install conduit (most likely it will need one of the heavier duty types of conduit such as Rigid Metal Conduit or PVC 80) and then run the wires through the conduit, or to replace the wires with an appropriately rated cable.
You should also clarify the meter "main service breaker" details. If that breaker is labeled and installed purely as a "service disconnect" then your "sub/main panel" is the true main panel. But if that "main service breaker" is acting as the "main panel" for your installation then you need to:
- Separate ground from neutral in your "sub/main panel" - which is then truly a subpanel.
- Connect the ground rods to that main breaker box (a.k.a., "meter main").
- Run a ground wire from the "main service breaker" to your subpanel. Meaning not 2/2/4 but rather 2/2/4/6, assuming this is 100A service.