It *seems* unhealthy because it IS unhealthy.
Right now it's working.
Soon it will stop, because you are starving your body and your brain- your muscles and skin.

Your metabolism is likely slowing to a crawl, and if you aren't experiencing some cognitive effects at this point, you will soon.
If you take this train far enough, you will end up very very sick, and in a place you can not get out of with out major intervention.

Do yourself a favour: Maybe get rid of the scale, only weigh yourself at the gym r doctor's office [preferably not more than once a week]

Maybe take the app off your mobile device and jot things down in a small note book during the day to put in at night.

600 calories is not even a resting metabolic intake that your body can sustain itself on without going into survival mode.

That is- a lot of people need anywhere from 1000 -1400 calories when they do nothing. If you are adding in exercise, you are creating such a large calorie deficit, that your body will at some point start trying to store all it can because it is responding as though it is starving.
Which it is.

By creating such a large calorie deficit each day, you slow your metabolism. Very low calorie diets such as the one you are talking about, can slow metabolism ten percent.

Bottom line:
You are not doing this in a healthy way.
Restrict your access to the measuring tools and come up with a healthy plan to exercise- please ask a doctor or close friend or mentor to assist you with this. I think having a second opinion may help you see what might be reasonable or not.

One more point:
You want to feel better about yourself and that is commendable.
But how much time are you now dedicating to dieting and exercise in general: And I mean thinking about what you are eating, are not eating, how many calories you "can" have, whether you did enough exercise or not.

What happens when you get on a scale and you haven't lost weight since the last time? I can't imagine that makes you feel very good, which makes some sense if you are trying to lose weight.

What doesn't make sense is that the time between weighing oneself may be hours.

All this focus and planning and restriction will eventually take over your life if you are not vigilant about getting a hold of it now.
Is food fun still? Sometimes we restrict and diet and guilt ourselves to the point where we feel BAD if we eat anything.

Which isn't right or healthy.