Your body can synthesize all the carbohydrates it needs through gluconeogenesis. There are no "essential carbs" the way there are essential fatty acids and amino acids. You can absolutely live on a diet of just fats and protein - and plenty of people do. The Inuit mostly did, and so do folks on carnivore diets today.
It might not be the healthiest long-term option, but it won't kill you from a lack of carbs. You're probably thinking of "rabbit starvation" which happens when you eat too much protein and not enough fat.