Understanding Pitch Detection with Autocorrelation Methods
For higher resolution periodicity estimation, given properly band-limited data, the data can be upsampled (interpolated) before autocorrelation, and/or, the autocorrelation results can be interpolated (Sinc kernel or some order of polynomial).  

This, of course, assumes the signal has a high enough S/N, and that the periodicity itself is unmodulated in timbre(harmonic content), phase, frequency, or amplitude, which isn’t always true of real-world pitched sounds.  Thus a larger number of periods may be needed for a some desired statistical error bound on the pitch estimation.