I think the "not eating enough" thing has been largely debunked, hasn't it? The issue is that you're likely to get bored, fall off the wagon, have insufficient energy for exercise, take in insufficient fat for your nervous system to function effectively, etc. Also, if you do really stick to an extremely low calorie diet long term, you're likely to become malnourished unless you are super careful about vitamins, minerals & your macros. People in famine hit regions, or with anorexia do just continue to lose weight. It's not healthy though!
If you go below a certain level of calories, nutrients etc, your body will start to prioritise certain processes. Hair falls out, skin appears less healthy, nails become weaker, your cognitive function will start to struggle. Fertility will temporarily cease - periods stop.
Unless you are very short, & naturally have a very light frame anyway, 600-800 calories per day is not really enough. You will lose weight, but it's going to be extremely difficult to maintain it without significant issues. And, when you hit your goal weight, you will struggle to maintain that as you will be used to a very low intake - and will likely have low muscle mass due to months of severe dieting. So won't be able to exercise to maintain on a "normal" diet.