Actually, since it is open to the public for business, it is not private property in the sense that a home is.
In your home, you do not have to entertain protected class individuals. In a business, such refusal would put you on the wrong side of a civil rights suit.
Personally, I think anybody who wants to deny my right of self defense needs to have a positive system in place to defend me. If not, I ignore such signs - in Missouri, that is my legal option, as signs only have force if somebody notices, then asks me to leave, and then I refuse.