8. (50 points) In a $4 n \times 4 n\left(n \in \mathbf{N}_{+}\right)$ real number table, the absolute value of each number does not exceed 1, and the sum of all numbers is 0. It is known that the absolute value of the sum of each row and the absolute value of the sum of each column are not less than $c$. Find the maximum possible value of $c$.

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8. $c_{\max }=2 n$.

Example: Take the number at the $i$-th row and $j$-th column of the number table as
$$
a_{i j}=\left\{\begin{array}{ll}
0, & 1 \leqslant i, j \leqslant 2 n \text { or } 2 n+1 \leqslant i, j \leqslant 4 n ; \\
1, & 1 \leqslant i \leqslant 2 n, 2 n+1 \leqslant j \leqslant 4 n ; \\
-1, & 2 n+1 \leqslant i \leqslant 4 n, 1 \leqslant j \leqslant 2 n .
\end{array}\right.
$$

Assume that the absolute value of the sum of numbers in each row and each column is greater than $2 n$.

Suppose there are $2 n+x$ columns with a positive sum (called positive sum columns), and $2 n-x$ columns with a negative sum (called negative sum columns), where $|x| \leqslant 2 n-1$; and $2 n+y$ rows with a positive sum (called positive sum rows), and $2 n-y$ rows with a negative sum (called negative sum rows), where $|y| \leqslant 2 n-1$.
Then the sum of all numbers in the positive sum columns
$$
u>2 n(2 n+x),
$$

and the absolute value of the sum of all numbers in the negative sum columns
$$
|-u|>2 n(2 n-x).
$$

Therefore, $u>2 n(2 n+|x|)$.
Similarly, the sum of all numbers in the positive sum rows
$$
v>2 n(2 n+|y|).
$$

Let $a$ be the sum of the numbers at the intersection of positive sum rows and positive sum columns, $b$ be the sum of the numbers at the intersection of positive sum columns and negative sum rows, and $c$ be the sum of the numbers at the intersection of negative sum columns and negative sum rows. Then
$$
\begin{array}{l}
a+b=u, b+c=-v \\
\Rightarrow a-c=u+v>2 n(4 n+|x|+|y|).
\end{array}
$$

On the other hand,
$$
\begin{array}{l}
a-c \leqslant(2 n+x)(2 n+y)+(2 n-x)(2 n-y) \\
=8 n^{2}+2 x y.
\end{array}
$$

Thus, $|x y| \geqslant x y>n(|x|+|y|)$, leading to
$$
(|x|-n)(|y|-n)>n^{2},
$$

which is a contradiction.