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Verbal Kint specified that \$G_0=\frac{a_0}{b_0}\$ and focused his attention upon on the roots. In the over-damped case (both numerator and denominator are way over-damped) the solutions are:

So, the transfer function can also be written out as:

Summary

Then perhaps you should refer to Verbal Kint's reference here. I don't want to provide, nor do I have the time or inclination to do so, a general approach for arbitrary order. I'm not sure one exists as I imagine the set of possible answers has often more than one item in it. I don't believe such a proof exists, but if you can prove that there is always one and only one possible answer in the set, I may yet give it a hack. But I'll need to see the mathematical proof before I bother.

Verbal Kint specified that \$G_0=\frac{a_0}{b_0}\$ and focused his attention upon on the roots. In the over-damped case (your numerator and denominator are both way over-damped) the solutions are:

In the over-damped case, both \$\omega_{_\text{H}}\$ and \$\omega_{_\text{L}}\$ are real-valued. So, the transfer function can also be written out as:

Relationships between \$\omega_{_\text{H}}\$, \$\omega_{_\text{L}}\$, and \$\zeta\$

For the over-damped situation, it turns out that \$\omega_{_\text{H}}\$ and \$\omega_{_\text{L}}\$ relate directly to \$\zeta\$:

$$\zeta=\frac12\frac{\omega_{_\text{L}}+\omega_{_\text{H}}}{\sqrt{\omega_{_\text{L}}\cdot \omega_{_\text{H}}}}=\frac12\frac{\omega_{_\text{L}}+\omega_{_\text{H}}}{\omega_{_0}}$$

It's pretty easy to see that as \$\omega_{_\text{L}}\to \omega_{_\text{H}}\$ then \$\zeta\to 1\$. But for \$\omega_{_\text{H}}\gg \omega_{_\text{L}}\$, then \$\zeta\to \frac12\sqrt{\omega_{_\text{H}}}\$.

It follows that wide bandwidth 2nd order filters tend to have large values for \$\zeta\$ (and therefore very small values for \$Q\$.) This fact helps in designing such filters. If \$\zeta\$ is large, then the design will tend towards using separate low-pass and high-pass filters, as separate sections that are concatenated. However, if \$\zeta\$ is small (\$Q\$ is larger), then the design will tend towards a unified bandpass design, instead.

There are a couple of interesting terms used to describe the fractional bandwidth of a system. One is \$B_{_\text{F}}=\frac{\omega_{_\text{H}}-\omega_{_\text{L}}}{\omega_{_0}}\$. The other is \$B_{_\text{F}}^{\:'} =2\frac{\omega_{_\text{H}}-\omega_{_\text{L}}}{\omega_{_\text{H}}+\omega_{_\text{L}}}\$. These are related to each other in this way: \$B_{_\text{F}}=\zeta\cdot B_{_\text{F}}^{\:'}\$. It turns out that \$0 \le B_{_\text{F}}^{\:'}\le 2\$ but that \$B_{_\text{F}}\to \sqrt{\frac{\omega_{_\text{H}}}{\omega_{_\text{L}}}}\$ for large ratios of \$\frac{\omega_{_\text{H}}}{\omega_{_\text{L}}}\$.

Just be aware that fractional bandwidth doesn't have a single definition in the literature.

Summary

Then perhaps you should refer to Verbal Kint's reference here. 

I don't want to provide, nor do I have the time or inclination to do so, a general approach for arbitrary order. I'm not sure one exists as I imagine the set of possible answers has often more than one item in it. I don't believe such a proof exists, but if you can prove that there is always one and only one possible answer in the set, I may yet give it a hack. But I'll need to see the mathematical proof before I bother.

Then perhaps you should refer to Verbal Kint's reference here. I don't want to provide, nor do I have the time or inclination to do so, a general approach for arbitrary order. I'm not sure one exists as I imagine the set of possible answers has often more than one item in it. But if you can prove that there is always one and only one possible answer in the set, I may yet give it a hack. But I'll need to see the mathematical proof before I bother.

Then perhaps you should refer to Verbal Kint's reference here. I don't want to provide, nor do I have the time or inclination to do so, a general approach for arbitrary order. I'm not sure one exists as I imagine the set of possible answers has often more than one item in it. I don't believe such a proof exists, but if you can prove that there is always one and only one possible answer in the set, I may yet give it a hack. But I'll need to see the mathematical proof before I bother.

Added Notes Per Comments from OP

If I understand correctly, this applies to voltage transfer functions.

No, it applies to all transfer functions. In voltage-in/voltage out cases the units are \$\frac{V}{V}\$, which can be treated as unitless. But in cases where the input units are current, for example, and the output is voltage then the gain value will carry units (Ohms, for example.)

However, my example is about the output impedance, described via transfer function H(s) (apologize if this is not the right nomenclature) which I want to find an equivalent circuit. I am not sure if active filter design relates to this.

Again, you can have a gain with units, such as Ohms. For example, suppose the following case:

^{simulate this circuit}

The transfer function is:

$$\begin{align*} G_s&=A\cdot\left[\frac{1}{\left(1+\frac{s}{\omega_{_\text{L}}}\right)\left(1+\frac{s}{\omega_{_\text{H}}}\right)}\right] \end{align*}$$

In this case, \$A=\frac{R_1\,R_4}{R_1 + R_2 + R_3 + R_4}\$ and the units are definitely \$\Omega\$.

I am not sure if active filter design relates to this.

The mathematics approach is similar. I'm not sure there's a problem. But I suppose you can identify one?

Does this apply to an output impedance as well?

Yes. As indicated above.

I don't mind the exact structure of the equivalent passive circuit, I just want to be able to draw it. Could be a ladder, shunt networks or similar.

Then perhaps you should refer to Verbal Kint's reference here. I don't want to provide, nor do I have the time or inclination to do so, a general approach for arbitrary order. I'm not sure one exists as I imagine the set of possible answers has often more than one item in it. But if you can prove that there is always one and only one possible answer in the set, I may yet give it a hack. But I'll need to see the mathematical proof before I bother.