$3 \cdot \frac{5}{77}$.
From $3a + 2b + c = 5, 2a + b - 3c = 1$, we get
$$\left\{ \begin{array}{l} 3a + 2b = 5 - c, \\ 2a + b = 1 + 3c \end{array} \Rightarrow \left\{ \begin{array}{l} 3a + 2b = 5 - c, \\ 4a + 2b = 2 + 6c \end{array} \right. \right.$$
Thus, $a = 7c - 3, b = 7 - 11c$.
Since $a, b, c$ are non-negative real numbers, we have
$$\left\{ \begin{array}{l} 7c - 3 \geqslant 0, \\ 7 - 11c \geqslant 0, \\ c \geqslant 0 \end{array} \Rightarrow \frac{3}{7} \leqslant c \leqslant \frac{7}{11}. \right.$$

Also, $m = 3a + b - 7c = 3c - 2$, so $-\frac{5}{7} \leqslant m \leqslant -\frac{1}{11}$.
Thus, $x = -\frac{5}{7}, y = -\frac{1}{11}$. Therefore, $xy = \frac{5}{77}$.