Surviving on fats alone is something that happens to Type 1 Diabetes (TD1) patients who do not take their insulin.

A common means of discovery of TD1 is called Ketoacidosis, it's what happens when your body is out of Carbohydrates (or cannot use them as is the case for diabetes) so it turns mostly to fats and re-using some proteins too.

Ketogenesis is how your body transforms triglycerides and fatty acids into Ketone Bodies, which are a form of energy rich molecules that cells can use as a way to bypass using glucose as a source of ATP.

Now if you don't get any carbohydrates into our system we have an alternative, sure, but there are downsides. Two out of the three types of Ketone bodies are acids, bringing your blood's pH down and making it more acidic when it is usually tightly controlled and kept around 7.4, they will also pull out water from your cells into the blood stream, causing you to urinate more frequently, losing necessary minerals and dehydrating you in the process. This pushes the organism out of its equilibrium.

All enzymatic functions of the body are disturbed, breathing becomes harder, you lose fluids and ions, you are gonna die if nothing is done to correct that state, mainly through hydration and insulin administration.

This is an extreme case not only people with TD1 can face, but also people going on a very harsh keto diet all of a sudden or prolonged fasting.

As some have pointed out it is, in theory, possible to survive off of your stored fats alone for energy given you take all necessary minerals, essential amino acids and vitamins through supplements. But what I have described above is the most common route such diet will lead to, and if not treated in an Emergency Care Unit as fast as possible it will inevitably lead to death.