Concatenation of language to itself zero times
$L^0$ is defined to be $\{\varepsilon\}$. (Since a string formed by concatenating zero strings together is the empty string).

This gives the answer to the first question - which is correct.

Now, $\{\varepsilon\}$ contains exactly one string - which is the empty string. Therefore: $|\{\varepsilon\}|=1$. (The given answer is wrong).

Note that $|\{\varepsilon\}|=1\ne0=|\varepsilon|$. Also, note that $\{\varepsilon\} \ne \emptyset$.