You’re right. It takes forever for the rabbit to reach the carrot. But the distance to the carrot is finite. It can be measured. The same is true of the length of a shoreline. We KNOW it can be measured, but if we keep defining it more and more granularly, eventually it becomes infinite.
There is a subtle difference in the analogy. The distance to the carrot is finite. The distance of the coast is infinite but the area encompassed by the coast is finite. Hence the definition of a fractional dimension or fractal.