When is a discrete time transfer function unrealizable?
You give a transfer function in $z^{-1}$, but for the purposes of making a state-space model, you need a transfer function in $z$ (more on that later).  So your $H$ becomes
$$H(z) = \frac{b_0 z + b_1}{z^2 + a_1 z + a_2}$$
When you invert that it's improper, which shows that you need to know information one time step into the future.  That's not possible (or, if it is, you should take up life as a stock broker).

(Note that you don't need state space to know this: you can just write the corresponding difference equation for your $H^{-1}$ and you'll see that it's asking for future values of the input to affect the current value of the output.)

Appendix

The reasoning for doing it in $z$ is because the time-domain state-space realization is

$$\begin{split} x_k = A x_{k-1} + B u_k \\ y_k = C x_{k-1} + D u_k\end{split}$$

For $D = 0$, the resulting transfer function is strictly proper in $z$ (but not in $z^{-1}$); for $D \ne 0$ the resulting transfer function in $z$ has equal orders in the numerator and denominator.