Your body only requires 1 tsp of glucose and it can be synthesized in the liver from protein (glucogenesis). Most tissues can be fueled with ketones, in fact some tissues prefer them to glucose. So you could survive eating no sugar (or carbohydrates) whatsoever. As evidence of this, look at someone fasting or on a hunger strike. They can survive about a month or so. Part of what keeps them alive is that the tiny amount of glucose they need can be created from proteins in their body. But that is also what will kill them, the body will catabolize the proteins from organs, like the heart.

People who live in artic environments often go many, many months without eating carbohydrates and they've managed to flourish for many generations in that environment.