[Proof] In $K$, each digit of $x$ and $y$ is 0 or 2, so each digit of $\frac{x}{2}$ and $\frac{y}{2}$ is 0 or 1, and thus each digit of $\frac{x}{2}+\frac{y}{2}$ in base 3 is 0, 1, or 2, and by
$$x \in[0,1], y \in[0,1]$$

we know
$$\frac{x}{2}+\frac{y}{2} \in[0,1] .$$

Conversely, for any number in $[0,1]$, each digit in base 3 is 0, 1, or 2, and it can clearly be written as the sum of two numbers whose digits in base 3 are 0 or 1. That is, it can be written in the form $\frac{x}{2}+\frac{y}{2}, x, y \in K$.

Therefore, we have
$$\left\{\left.\frac{x}{2}+\frac{y}{2} \right\rvert\, x, y \in K\right\}=[0,1],$$

hence
$$S=\{x+y \mid x, y \in K\}=[0,2]$$