Strictly speaking, this is not a double negative; I believe this is known as negative concord. Basically, if the sentence is in negative, every word like anything, anywhere, etc. should be negated too.
An example of a true double negative is «не могу не заметить» (i.e. “I cannot not remark”), where the double negative collapses to a positive, just like in logic.
The sentence translated: У него ничего нет = He doesn't have anything/He has nothing.
In other words, consider ничего нет as the doesn't have anything form.
Your issue comes from the fact that you would like to write He has nothing with a single word. Unfortunately, Russians just stick to ничего нет for that.
Sometimes, these kinds of phrases are genuinely illogical and I wonder why people don't just fix the illogical error with the language rather than persist. Example: Translating this in English essentially means "they not have nothing" but not having nothing means they possess something, ergo, I now believe they have something. I notice there are all of these nuances too, rules that aren't actually rules because of all of the examples where you can break them. I wonder if man-made (as in purpose-built) languages like Epseranto pull those stunts too?