Why is a linear passive circuit stable, i.e. why does its impulse response approach zero over time?

In a circuit that is composed of only Ls, Cs and Rs ...

An input impulse will store energy in the Ls and Cs. The stored energy will be dissipated in the Rs, and will tend to zero over time.

Conversely, if there are no Rs, no means of dissipation, then the energy will remain stored, and the impulse response will last indefinitely.

I suspect the problem may arrive from the requirement of an impossible domain for passive component values. Your characteristic equation breaks down (as I'm sure you already know) into: \$\left(s+1\right)\left(s^2+s+1\right)\left(s^2-s+1\right)\$. The first two factors certainly can be formed with passive components. But the last term appears to require a negative-valued component.

Can you can find any passive circuit arrangement where the characteristic equation is \$s^2-s+1\$?

The simple answer is related to the Routh-Hurwitz stability criterion. This means that it simply doesn't matter how the terms of the polynomial end up as long as all the roots have negative real part. Only by satisfying this criterion, the impulse response will decay in time.