Stability is when a situation is at a relative free energy minimum OR when there is no possible or reasonable mechanism for something to happen. An historical example: nitroglycerine was a favored explosive in mining etc. Nitro, undisturbed, was stable; but if you shook it or jarred it too hard it exploded, severely damaging the person who shook it; Along comes Alfred Nobel; he figures a way to stabilize nitro and invents dynamite; dynamite sticks can be dropped They are stable! or are they? The profits started the Nobel prizes.
Most compounds are stable if in isolation including strong bases or acids; some compounds are inherently unstable, an example being Pure H2O molecules. They cannot exist, for very long at least, without self-ionizing to H3O+ and OH- ions. So the combination of a strong base mixed with a very weak acid is a mix that might be unstable ie not at equilibrium. It might react if a mechanism is possible and if the base and or acid are strong enough for a reaction.
Stability is an unfortunate word to describe reactivity. A base such as OH- is more reactive than H2O when reacting with acids. Stability depends on the conditions and displacement from equilibrium.