Stability of passive networks
Yes, an RLC network of passive components will be stable, because they only delay or absorb power.
A direct effect of BIBO stability is that the transfer function is absolutely integrable:
$$ \int_0^\infty \biggl|h(t)\biggr| \, dt \lt \infty $$
Which for any real load (finite impedance) means that the energy output is finite for any finite input.  Lyapunov stability relies heavily on the idea of energy transfer, to great effect even in non-linear systems.
With a network of passives, there is no energy source other than the input; with any resistance at all, the output energy will always be less than the input energy.
Stability generally becomes a concern with feedback.  For instance an op-amp circuit with positive feedback will eventually max out its output.