Not necessarily. If you assume an ideal ruler and compass, it‘s pretty easy to construct the square root of any number which means that you can easily put a dot on that number line which is provably not a rational number. There is the question of whether continuity actually exists (are time and space quantized like matter and energy or are they continuous? this is currently unknown) and the fact that your paper and the line on it are composed of discrete molecules. But if the real numbers are, in fact, real, then the probability that your dot is at a rational point is actually 0 since while the number of rationals in [0,1] is infinite, it’s only countably infinite and the number of irrationals in [0,1] is uncountably infinite meaning |ℚ|/|ℝ\ℚ|=0.