The scaled lengths of molecular trajectories obtained by molecular dynamics simulation of a hard-sphere fluid are shown to have the same fractal dimension D = 2 as the random walk. Self-similarity first appears on length scales typically a factor of 25 greater than the mean free-path length, whereas for the simple random walk with constant step size the onset occurs after only six steps; the reason for the slow convergence is shown to be the near exponential distribution of intercollision path lengths of the fluid molecules. The influence of density on the scaled path lengths is also discussed.