We study spherically symmetric oscillations of electrons in plasma in the frame of classical electrodynamics. Firstly, we analyze the electromagnetic potentials for the system of radially oscillating charged particles. Secondly, we consider both free and forced spherically symmetric oscillations of electrons. Finally, we discuss the interaction between radially oscillating electrons through the exchange of ion acoustic waves. It is found that the effective potential of this interaction can be attractive and can transcend the Debye-Hückel potential. We suggest that oscillating electrons can form bound states at the initial stages of the spherical plasma structure evolution. The possible applications of the obtained results for the theory of natural plasmoids are examined.