Write down the given function for the drone's height.

The height of the drone above the ground after $t$ seconds is given by the function:

$$h(t) = t^2 - 6t + 8$$

To find the height at $t = 4$ seconds, replace every $t$ in the function with 4:

$$h(4) = (4)^2 - 6(4) + 8$$

First, evaluate the square and the multiplication:

$$h(4) = 16 - 24 + 8$$

Next, perform the subtraction and addition:

$$h(4) = -8 + 8$$ $$h(4) = 0$$

The height of the drone at $t = 4$ seconds is $\boxed{0 \text{ units}}$.