Are you allowed to cover a water heater in California?

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what sort of water heater?  plumbing code tells you what clearance is required (which may depend on type of water heater) ... you can ask for free at your local building permits [government] office.  -- grampa

If it is electric you can build a small room around it with a door.  Make sure you make it big enough so when the water heater goes bad you can get it out the door to replace it.   To be safe,  put a label on the door of what is inside.   If it is gas DO NOT build a room around it.   You want good air flow around it in case there is a gas leak and so that exhaust is properly vented.    Consider getting a tankless water heater.   They are a LOT smaller and just hang on the wall.  

There is a minimum spacing around the water heater, but having on in some type of closet is very common.

It's very common to have them like that.  Checking your local building codes would tell you for sure.  They often vary from state laws.  You'd want to make sure there is enough space around it and also ensure that you could get it out (and a new one in), including dealing with connections, when it hits the end of life.  

You mean to build a "house" around it (more like a shed, or a closet)?  I do not see why not.  I do remember seeing a lot of water heaters out in the weather in the poor parts of LA. I am not sure why.