As far as I know, legal definitions don't require a closet. However, I think any bedroom should have one. Certainly, if you are asking an appraiser to consider your place a two bedroom, I think you are better off adding a closet. And public records will still show the property as a one bedroom unit until you change that. Regarding the size, I didn't find a size requirement in SF or California, but other states require 60 square feet.
I see listings all the time where non-closeted fully enclosed rooms are called bedrooms. I generally feel like this is a little misleading - even if the owners happily use the room as a bedroom. I think a closet is essential.
Have you chosen your appraiser already? I would suggest talking to that person regarding their requirements. I can make a referral if you wish.