5. Set
$$
M=\left\{(x, y) \| \log _{4} x+\log _{4} y \leqslant 1, x, y \in \mathbf{N}_{+}\right.
$$

has ( ) subsets.
(A) 4
(B) 16
(C) 64
(D) 256
5.D.

From $\log _{4} x+\log _{4} y \leqslant 1$, we get $1 \leqslant x y \leqslant 4$.
Thus, $(x, y)=(1,1),(1,2),(1,3),(1,4)$,
$$
(2,1),(2,2),(3,1),(4,1) \text {. }
$$

Therefore, the set $M$ has $2^{8}=256$ subsets.