We can write $ |z|^2 = z \overline{z}, $ so the equation becomes \[z^3 + z \overline{z} + z = 0.\]Since $ a $ and $ b $ are positive, $ z = a + bi $ is nonzero. Thus, we can divide both sides of the equation above by $ z, $ which gives us \[z^2 + \overline{z} + 1 = 0.\]Then $ (a + bi)^2 + \overline{a + bi} + 1 = 0, $ or \[a^2 + 2abi - b^2 + a - bi + 1 = 0.\]Equating real and imaginary parts, we get \begin{align*} a^2 - b^2 + a + 1 &=0, \\ 2ab - b &= 0.\end{align*}From the second equation, $ b(2a - 1) = 0 $. Since $ b $ is positive, $ 2a - 1 = 0, $ so $ a = \frac{1}{2} $. Then from the first equation, \[b^2 = a^2 + a + 1 = \frac{7}{4}.\]Since $ b $ is positive, $ b = \frac{\sqrt{7}}{2} $. Thus, $ (a,b) = \boxed{\left( \frac{1}{2}, \frac{\sqrt{7}}{2} \right)} $.