How to make an objects variables private?

The easiest thing to do without having to mess with metatables is to just keep the variables outside of the object. I know this goes against everything you learned about OOP, but it's what works the best in Lua, which isn't OOP. Example:

local Pizza = {}

    function Pizza.new(...)
        local slices = 8
        local toppings = {...}

        local pizza = {}

        function pizza:eat(x)
            local originalSlices = slices
            if slices > 0 then
                slices = math.max(0, slices - (x or 1))
            end

            return originalSlices - slices--returns slices eaten
        end

        --[[
            What are you trying to access pizza.slices directly for? That's the exact opposite of what OOP is. In OOP languages, you'll always see x = pizza.GetX() instead of x = pizza.x.
        ]]
        function pizza:getSlices()
            return slices
        end

        function pizza:addTopping(toping)
            table.insert(topings, toping)
        end     

        return pizza
    end

return Pizza

'local' is the same thing as making a private value in Java.

So

local cake = 10
do
    print(cake)
end

Output:

nil

In Lua values are local to the scope of code that they're in. So creating a value outside of a scope will make it not exist in said scope. If you don't put "local" behind the value name then it is a public value.

In your case, your code should look like this:

do

local Pizza = {}

    function Pizza.new(...)
     return setmetatable({
           Slices = 8;
            Toppings = {...};
         Cooked = false;
      }, Pizza)
    end

    Pizza.__index = Pizza

    function Pizza:eat(x)
      if x ~= nil then
          self.Slices = self.Slices - x;
      else
            self.Slices = self.Slices - 1;
       end
    end

    function Pizza:addTopping(x)
        table.insert(self.Toppings, x);
    end

    Pizza.__newindex = function()
       error("Pizza object is private");
    end



    return Pizza

end

----

--This is outside of the scope, it will return a error.
pepperoni = Pizza.new("Pepperoni");
print(pepperoni.Slices);
pepperoni.Slices = 2;
print(pepperoni.Slices);
pepperoni.cheese = 5;

Though your problem is irrelevant: if you want your data to be protected just turn on FilteringEnabled.