Do you need to eat carbohydrates to fuel your body? No.
Carbohydrates (sugars and starches) are digested in the intestinal tract and transformed into a molecule known as glucose; this is the substance commonly referred to as blood sugar.
Glucose is utilized as a energy source primarily for the brain and a small amount for muscle metabolism. The remainder of the body primarily burns molecules known as fatty acids for energy. After absorption, some of the glucose is used immediately to fuel the brain and muscle. What is not needed is converted into fatty acids and stored in the fat cells or converted into glycogen and stored in muscle and liver tissue.
Hypothetically, if a person were to completely refrain from eating any carbohydrates (the starches and sugars that are transformed into glucose), our bodies have the ability to produce all the necessary glucose from stored glycogen and fatty acids. We are so efficient at doing this that our bodies can maintain very normal levels of blood sugar for extended periods of time (days to weeks) without ever consuming a single gram of carbohydrates.
If you believe you are experiencing occasional symptoms of low blood sugar that improving after eating, I suggest you read my post about Tums.
This being said, I am not advocating that you should stop eating carbohydrates completely. My point is debunk the myth that you somehow need to keep eating carbohydrates through the day to fuel your system.