5. If the sum of all elements in a non-empty subset of $\{1,2, \cdots, 9\}$ is a multiple of 3, the subset is called a "Jin state subset". Then the number of such Jin state subsets is $\qquad$ .
5. 175.

If a proper subset is a peculiar subset, then its complement is also a peculiar subset. Therefore, we only need to consider peculiar subsets with fewer than or equal to 4 elements.

Thus, there are 3 peculiar subsets with only 1 element; there are $\mathrm{C}_{3}^{2}+3 \times 3=12$ peculiar subsets with exactly 2 elements; there are $3+3 \times 3 \times 3=30$ peculiar subsets with exactly 3 elements.

For 4-element subsets, since the remainders of the 4 elements when divided by 3 must have two that are the same, it is known that there are only
$$
(0,0,1,2),(1,1,1,0),(1,1,2,2),(2,2,2,0)
$$

these four possible combinations of remainders.
Thus, the number of peculiar subsets with 4 elements is
$$
\mathrm{C}_{3}^{2} \mathrm{C}_{3}^{1} \mathrm{C}_{3}^{1}+2 \mathrm{C}_{3}^{3} \mathrm{C}_{3}^{1}+\mathrm{C}_{3}^{2} \mathrm{C}_{3}^{2}=42 \text {. }
$$

Since the universal set is also a peculiar subset, the number of peculiar subsets is
$$
2(3+12+30+42)+1=175 .
$$