Is there such thing as white wheat berries?

Sorry, my answer was incorrect (it used to say that there is no white wheat). The only defense I have is that such newfangled things like white wheat have not yet reached my part of the world (is Europe really that old-fashioned?). Big thank you to Jefromi for correcting me and teaching me something new.

"Normal" white flour is, as I described in the old answer, made from the inner part of the wheat berry, discarding the yellow-brownish hull. You can't make this at home. But it turns out that there is such a thing as white wheat. When it is used to produce whole flour, the result is a flour which is white in color (but behaves more like whole flour when baked, and has nutritional properties like whole flour). So if your definition for white bread is just based on color, it is probably possible to make white flour from white wheat. They say that the taste is milder than "yellow" whole flour, too, closer to the refined white flour. But from what I read, you probably won't be able to produce a fluffy soft baguette with it, just because it bakes differently.

If you want to make flour which is equivalent to the common white flour in the supermarket, you are still out of luck. The machines for removing the bran are too complicated, you can't do it in a grinder (look here for a patent for a bran removing device). But if "white whole wheat flour" is good enough for your purposes, I don't see a reason why you can't grind albino wheat at home.

I use hard white wheat as well as hard red wheat for a lot of things.

When you think of the hard, almost astringent, taste of whole wheat it is almost certainly red wheat.

White wheat is much milder (and as you would expect, paler in color) but, unlike soft southern wheats, still has enough protein for bread doughs. The bread made from finely ground white flour is much closer in consistency and flavor to white bread. I can also use it in my normal bread recipes and it doesn't require extra sweeteners or the extra protein that red wheat requires to make light bread. I have started using hard white flour in all my baked goods and use the harsher red wheat when I make bulgar wheat, sprouts, or just hot cereal.

As Rumtscho's answer says- The flour from whole hard white wheat still has the bran and so is not "white flour" but I think you will find that it still sufficient for your purpose.

As for obtaining them: Hard wheats are from the north and here in Texas they have to be shipped. Even so, all the places that I frequent for bulk wheat berries carry red as well as white. I can't make recommendations because they are local bulk food/food storage places.