This paper is concerned with volume-constrained minimization problems derived from Gamow's liquid drop model for the atomic nucleus, involving the competition of a perimeter term and repulsive nonlocal potentials. We consider potentials given by general radial nonnegative and nonincreasing kernels which are integrable on $\mathbb{R}^n$ , such as Bessel potentials. We show that, under reasonable regularity assumptions and a bound on the first moment of such kernels, the problems admit minimizers of arbitrarily large mass. In addition, we prove that, up to renormalization, any sequence of minimizers converges to the ball as the mass goes to infinity. Finally, we study the stability of large balls, and show that our bound on the first moment of the kernels is a sharp stability threshold for large balls. A direct consequence of the instability of large balls above this threshold is that there exist nontrivial compactly supported kernels for which the problems admit minimizers which are not balls, that is, symmetry breaking occurs.