Extending recent numerical studies on two-dimensional amorphous bodies, we characterize the approach of the elastic continuum limit in three-dimensional (weakly polydisperse) Lennard-Jones systems. While performing a systematic finite-size analysis (for two different quench protocols), we investigate the nonaffine displacement field under external strain, the linear response to an external δ force, and the low-frequency harmonic eigenmodes and their density distribution. Qualitatively similar behavior is found as in two dimensions: The classical elasticity description breaks down below a surprisingly large length scale ξ, which in our system is approximately 23 molecular sizes. This length characterizes the correlations of the nonaffine displacement field, the self-averaging of external noise with distance from the source, and gives the lower wavelength bound for the applicability of the classical eigenfrequency calculations. Moreover, we demonstrate that the position of the “Boson peak” in the density of vibrational states is related to this self-averaging length ξ.