【Analysis】First, take the set
$$
S=\left\{2^{i} 3^{2011-i} 10 \leqslant i \leqslant 2011, i \in \mathbf{Z}\right\} .
$$

Then, except for $2^{2011}$ and $3^{2011}$, the other 2010 elements in set $S$ are all good elements.

Next, we prove that set $S$ can have at most 2010 good elements.

Let set $S=\left\{a_{1}, a_{2}, \cdots, a_{2012}\right\}$, and the set of prime factors of these 2012 elements is
$$
P=\left\{p_{1}, p_{2}, \cdots, p_{n}\right\} .
$$

For any $a \in \mathbf{Z}_{+}$ and prime $p$, let $d(a, p)$ be the exponent of prime $p$ in the prime factorization of $a$.
Assume $a_{1}=\max \left\{a_{1}, a_{2}, \cdots, a_{2012}\right\}$.
Then, for any $y, z \in S$, we have $y z < a_{1}^{2}$.
Thus, $a_{1}^{2} \nmid y z$.
Therefore, $a_{1}$ is not a good element.
Since no element in set $S$ is a multiple of another, in the prime factorization of all $a_{i}$ in $S$, $a_{1}$ must have a prime $p_{k}$ whose exponent is not the largest.

Let the largest element among those with the largest exponent of $p_{k}$ be $a_{2}$. Then we can prove that $a_{2}$ is also not a good element.
Indeed, for any $a_{i}, a_{j} \in S \backslash\left\{a_{2}\right\}$, we have
$$
\begin{array}{l}
d\left(a_{i}, p_{k}\right) \leqslant d\left(a_{2}, p_{k}\right), \\
d\left(a_{j}, p_{k}\right) \leqslant d\left(a_{2}, p_{k}\right) . \\
\text { If } d\left(a_{i}, p_{k}\right)+d\left(a_{j}, p_{k}\right)<2 d\left(a_{2}, p_{k}\right) \text {, then } \\
a_{2}^{2} \nmid a_{i} a_{j} ; \\
\text { If } d\left(a_{i}, p_{k}\right)+d\left(a_{j}, p_{k}\right)=2 d\left(a_{2}, p_{k}\right) \text {, then } \\
d\left(a_{i}, p_{k}\right)=d\left(a_{j}, p_{k}\right)=d\left(a_{2}, p_{k}\right) \\
\Rightarrow a_{i}<a_{2}, a_{j}<a_{2} \\
\Rightarrow a_{2}^{2}+a_{i} a_{j} .
\end{array}
$$

Thus, $a_{2}$ is also not a good element.
In summary, set $S$ can have at most 2010 good elements.
Example 3 Let set $S$ have 2012 elements, satisfying that the ratio of any two elements is not an integer. For any element $x$ in $S$, if there exist distinct elements $y, z$ in $S$ different from $x$, such that $x^{2} \mid y z$, then $x$ is called a "good element". Find the maximum possible number of good elements in $S$.