The given information implies that $a + b + c = 0$ and $a^2 + b^2 + c^2 = 30$. Then
$$0 = (a + b + c)^2 = a^2 + b^2 + c^2 + 2ab + 2ac + 2bc = 30 + 2(ab + ac + bc).$$
Therefore $2(ab + ac + bc) = -30$ and the requested arithmetic mean is $\frac{ab+ac+bc}{3} = \frac{-15}{3} = -5$.

Consider the system of equations implied by the conditions of the problem,
$$\begin{aligned} a + b + c &= 0 \\ a^2 + b^2 + c^2 &= 30, \end{aligned}$$
and suppose that $a = 0$. Then $b + c = 0$, so $b = -c$, and substituting into the second equation gives $2b^2 = 30$, from which $b = \pm\sqrt{15}$ and $c = \mp\sqrt{15}$. If one assumes that the requested arithmetic mean is determined by the given information, independent of the value of $a$, then
$$\frac{ab + ac + bc}{3} = \frac{0 + 0 - \sqrt{15} \cdot \sqrt{15}}{3} = -5.$$