R. Feynman was troubled by this question. He claimed there was no classical explanation. In fact by his classical analysis, the electron would spiral into the proton in approximately one nanosecond.This problem is formally the referred to as "atomic stability", in this case of "the atomic stability of the hydrogen atom". In 1911 Planck proposed his second quantitation theory where he introduced the concept of an energy sea that we now refer to as Zero-point radiation. after a rocky start. it is now mainstream research that is used to in QED, "Lamb shift", the "anomalous g-factor of the electron" and "atomic stability of hydrogen" I suggest a look at the following link as background: https://en.wikipedia.org/wiki/Zero-point_energy
No, there is no classical explanation of atom/matter stability. Classical physics (a combination of Newtonian mechanics + electrodynamics - Larmor) predicts the total loss of energy of the electron in orbit by radiation, therefore the "classical elementary atom" proposed by E. Rutherford in 1911 would not exist. So the question in the title has a very simple almost trivial answer.
If there was a classical explanation for the stability of the Hydrogen atom there would be no reason for a quantum mechanical description. Classical mechanics simply cannot provide explanations for several observed phenomenon and theoretical descriptions of atomic systems.
In short, classical electrodynamics predicts that an accelerating charge will emit radiation. Conservation of energy states that if energy appears in the radiation field, the accelerating charge must lose mechanical energy. Along with other reasonable assumptions there is no other conclusion than the particle orbit gets smaller, eventually collapsing on the other particle.
There are a lot of things one can question to try and fix this. For example, both the electron and proton are moving, as current loops, about the center of mass. Could they be exchanging classical radiation? I don't think so. People have tried for quite some time to look for issues with the early classical descriptions of atoms to see if adding more physics would lead to a stable state. This has failed. Much like Mercury's orbit not following Newton's law of motion using his gravity model. Eventually Einstein had to develop General Relativity to describe this.
This inability to describe the stable atom with classical physics, and the ability to do so with QM, is one of the major achievements of QM.
There is no classical explanation for it for a lot of reasons, one of them being that in classical physics (and in chemistry, etc.) words like electrons, atoms, etc are just that: words, placeholders, wildcards...
In classical physics, an electron is defined as the theoretical smallest part of a charge or as a « grain » of charge or as the particle making up electrical currents, charges, etc.
That kind of macroscopic definition says nothing about the behavior of a single electron.
Therefore, the popular / classical image of the electron as a very tiny spherical ball of electricity is just plain wrong. That’s why it leads to apparent paradoxes.
The closest thing to a classical explanation is a calculation that indicates that electron orbitals in a hydrogen atom correspond to resonant 3D standing wave patterns, with the wave packet associated with an electron being bounded by the E field from the nucleus. Though some of the patterns do have a non-zero amplitude at the position of the nucleus, the probability of an interaction within that very small volume is quite low. There are some unstable nuclei that have the ability to do a “K capture”.