The minimum measurable velocity will vary with the type of instrument and configuration, and there are a few factors that will dictate the quality of the data (SNR, for example). The below answer is meant as a theoretical approach to the question.

Speed/Velocity

First off a reminder about the difference between speed and velocity, and how these two parameters relate to statistical uncertainty. We distinguish between the term velocity (which involves direction of motion aswell as the rate) and speed, which only involves the rate and not direction. 

Speed is the absolute value of the velocity:

If we have a velocity component U, then the mean:

<U> = sum(U)/N

U = single ping velocity component
<U> = mean
N = number of pings
sig = standard deviation (gives a measure of how spread out the variance of the mean is. The smaller the standard deviation, the more of the variance is concentrated close to the mean)

That is; the mean is an arithmetic mean and the uncertainty in the mean is the uncertainty of each separate measurement divided by number of measurements. That means that the uncertainty in the velocity measurement gets smaller and smaller towards zero as long as one continue to measure. Therefore, there is no limit to how low the measurement uncertainty you can get.

If we work with speed, then: (ref. Pytagoras)

speed = sqrt(U^2 + V^2 +eps),

where

The uncertainty will get very low if we first take the mean of U and V. Doing that we can get good speed estimates. If, instead, one takes the mean of each speed measurement, we will not get rid of the uncertainty and speed will not converge towards zero even though both velocity component does. Depending on instrument setup, the sig(U) and sig(V) can be relatively large so that the error may be larger than one initially would expect.

Regarding the definition of accuracy (ref. instrument specific data sheet):

Direction 

The problem with direction is related to the above. Direction is normally defined as:

dir = 90-atan2(V/U)

(check out the wikipedia article for a good description of atan2)

The current direction measurements will be spread around in all directions when U moves toward zero, so one needs to make sure to find the mean velocity components first and then compute mean direction as a second step.