Instead of looking at it as being unhealthy for not having hobbies, let’s look at it as “what you’re missing out on by not having any hobbies.”
Believe it or not, being more busy actually makes you get stuff done ‘better.’ By increasing the amount of things that you do, you are more limited on time, so you then have to finish the other things instead of procrastinating.
Making yourself busy actually seems to create more time.
By having an enjoyable hobby, you will have an escape from life to go do what you love to do. For me, it is lifting weights. I love going from my hard classes to the gym—I feel such a large amount of stress lifted (nice) off my shoulders.
A lot of people struggle with finding value in their own life. People don’t have anything unique to their name—just another average joe. By learning a cool skill like playing the piano or even creating a nice garden in your backyard, you have something that can give your life value/make you unique.
There is just a few reasons why having a hobby is good, so the direct answer to your question would be that missing out on all the above is how unhealthy it is.