Please see, like an atom that has a ground state, and in that state it would remain always if it not disturbed by some influence from outside, (some irradiation with other particles), so there are nuclei whose state is the ground state. Such nuclei have no excess of energy to get rid of.
Let me give some examples: there are nuclei with too many protons, s.t. teir configuration is unstable. They are likely to undergo $\beta ^+$ decay, and emit a positron. Other nuclei have to many neutrons, s.t. they may emit one or more neutrons, or may undergo $\beta ^-$ decay.
In short, yes, there are stable nuclei, or, in other words, there are ground states for nuclei.
Long lived unstable isotopes, are not in the ground states, but in some resonant states, very narrow from the point of view of the energy range encompassed by that resonance. The narrower the resonance, the longer is its half-life. But, I repeat, a resonant state, long lived as it may be, is not a ground state.