Is a naturally radioactive elements on its ground state? Ground state is defined as the zero energy state. No energy can be taken from it. Still, radioactive elements can emit radiation. Do we consider them to be in a ground state?
 A: Unstable nuclei are not in their ground state, but then, it's not clear that stable nuclei exist, at all. 
We are currently searching very hard for proton decay: https://en.wikipedia.org/wiki/Proton_decay. A simple model has the proton decaying into a positron and two gammas. The positrons could then annihilate with the remaining electrons, leaving nothing but gammas at the end. Should it be found, then we will know that all matter that we can see around us is essentially just a metastable state and the ground state is radiation.
A: They are metastable states, as you can see here:

A: A radioactive nucleus is not  in its ground state of course.   The state may be unstable, or metastable (which implies a slow decay).  Of natural elements, only iron-56 nuclei and smaller have lower energy than their fission
products.  This means that a LOT of nuclei are metastable (though the half-life might be VERY long) with respect to some kind of fission.
Light nuclei can go to lower energy by fusion, but that requires a second particle to combine with, so it's hard to call them 'unstable'.
Very heavy nuclei, including such items as neutron stars, are theoretically possible,
but I don't know much about their stability.
