I actually like the repetition as I think it’s effective

As a native speaker of English, it seems completely acceptable as a sentence. However, I personally would probably add an expression to intensify it. I might say, “I am tired of just being tired all the time.”

It’s a great sentence. The real question is whether it means what you think it does!

It's best to avoid using the same word twice, so you could say "I'm fed up with being tired"

It sounds natural to me and really drives the point across.