5. Given that $A$ and $B$ are two distinct points on the parabola $y=3-x^{2}$ that are symmetric with respect to the line $x+y=0$. Then $|A B|$ equals ( ).
(A) 3
(B) 4
(C) $3 \sqrt{2}$
(D) $4 \sqrt{2}$
5. C.

Since points $A$ and $B$ are symmetric with respect to the line $x+y=0$, let point $A(a, b)$ and $B(-b, -a)$.
Also, points $A$ and $B$ lie on the parabola $y=3-x^{2}$, so
$$
\begin{array}{l}
\left\{\begin{array}{l}
b=3-a^{2}, \\
-a=3-(-b)^{2}
\end{array}\right. \\
\Rightarrow\left\{\begin{array} { l } 
{ a = - 2 , } \\
{ b = - 1 }
\end{array} \text { or } \left\{\begin{array}{l}
a=1, \\
b=2 .
\end{array}\right.\right.
\end{array}
$$

Without loss of generality, let point $A(-2, -1)$ and $B(1, 2)$. Then
$$
|A B|=3 \sqrt{2} .
$$