Is it natural to say the following?
Let's say you just got a phone call, and you want to tell the caller what you are doing at the moment.
Your first option
I am having my hair cut.
is natural and grammatical. The following are commonly used too (See this Ngram).
"I am getting my hair cut", "I am having a haircut", "I am having my hair done", and "I am getting a haircut"
As Ronald Sole said in the comment, you could say it in the passive voice: "My hair is being cut". I personally think the passive form (saying this exact sentence over the phone to the caller) is better suited for a situation when you are in agony - you don't want to have your hair cut, but you are being made to.
Your second option "I have my hair being cut" is not idiomatic (at least I have not heard anyone say that).