8. Let $\left\{\begin{array}{l}\sin \alpha=\sin (\alpha+\beta+\gamma)+1, \\ \sin \beta=3 \sin (\alpha+\beta+\gamma)+2, \\ \sin \gamma=5 \sin (\alpha+\beta+\gamma)+3 .\end{array}\right.$

Then the possible values of $\sin \alpha \cdot \sin \beta \cdot \sin \gamma$ are
8. $\frac{3}{64}, \frac{1}{8}$.

From the conditions, we have
$$
\begin{array}{l}
\sin \alpha+\sin \beta=4 \sin (\alpha+\beta+\gamma)+3 \\
=\sin \gamma-\sin (\alpha+\beta+\gamma) \\
\Rightarrow 2 \sin \frac{\alpha+\beta}{2} \cdot \cos \frac{\alpha-\beta}{2} \\
=-2 \sin \frac{\alpha+\beta}{2} \cdot \cos \frac{\alpha+\beta+2 \gamma}{2} \\
\Rightarrow 4 \sin \frac{\alpha+\beta}{2} \cdot \cos \frac{\alpha+\gamma}{2} \cdot \cos \frac{\beta+\gamma}{2}=0 .
\end{array}
$$
(1) When $\sin \frac{\alpha+\beta}{2}=0$,
$$
\alpha+\beta=2 k \pi(k \in \mathbf{Z}) .
$$

Substituting into the third equation of the system, we get
$\sin \gamma=\sin (\alpha+\beta+\gamma)=-\frac{3}{4}$
$\Rightarrow \sin \alpha=\frac{1}{4}, \sin \beta=-\frac{1}{4}$.
Then $\sin \alpha \cdot \sin \beta \cdot \sin \gamma=\frac{3}{64}$, when $\alpha=\arcsin \frac{1}{4}$, $\beta=-\arcsin \frac{1}{4}, \gamma=-\arcsin \frac{3}{4}$, it can be achieved.
(2) When $\cos \frac{\alpha+\gamma}{2}=0$,
$$
\alpha+\gamma=2 k \pi+\pi(k \in \mathbf{Z}) .
$$

Substituting into the second equation of the system, we get
$$
\begin{array}{l}
\sin (\alpha+\beta+\gamma)=-\sin \beta=-\frac{1}{2} \\
\Rightarrow \sin \alpha=\sin \gamma=\frac{1}{2} .
\end{array}
$$

Then $\sin \alpha \cdot \sin \beta \cdot \sin \gamma=\frac{1}{8}$, when $\alpha=\beta=\frac{\pi}{6}$, $\gamma=\frac{5 \pi}{6}$, it can be achieved.
(3) When $\cos \frac{\beta+\gamma}{2}=0$,
$$
\beta+\gamma=2 k \pi+\pi(k \in \mathbf{Z}) .
$$

Substituting into the first equation of the system, we get
$$
\begin{array}{l}
\sin (\alpha+\beta+\gamma)=-\sin \alpha=-\frac{1}{2} \\
\Rightarrow \sin \beta=\sin \gamma=\frac{1}{2} .
\end{array}
$$

The discussion is similar to (2).
In summary, the possible values of $\sin \alpha \cdot \sin \beta \cdot \sin \gamma$ are $\frac{3}{64}$ and $\frac{1}{8}$.