Centrifugal force seemed to be the answer.
Think of a rod of iron and a strong magnet. If these two objects are close enough they will attract one another strongly and come together with a crash!
But join them with a piece of string, and then spin the magnet round and round the iron rod. The outward force, called the centrifugal force, of the magnet as it moves in its circular path compensates for the equal and opposite force of attraction between the magnet and the iron rod, so the two objects stay together, but separate.
This is how and why the earth and the sun do not collapse together into one single molten mass. Despite the very strong gravitational force attracting the earth downwards towards fusion with the huge sun, the fact that the earth is circulating around and around the sun generates the outward centrifugal that keeps us apart (at least for as long as we are going to be alive!).
So Rutherford and others, including Hanataro Nagaoka, thought that electrons must be moving very rapidly in circular orbits thus using the centrifugal force of their motion to prevent them from being fatally attracted down and down into a massive meeting with the protons at the center.
Nagaoka even pictured the electrons behaving like little planets and just like our solar system, each electron would have its own orbit (so it did not crash into other electrons in other orbits!), and that the chemical properties of such atoms could then be related to the nature, spacing and characteristics of these electron orbits.
There was only one problem with this picture; electrons are not planets. If a negatively charged electron really moved in a circular obit in this way it would be constantly changing in direction (towards the center - actually), and thus "accelerating", in the classic meaning of this word.
An accelerating charged particle must be constantly changing in energy and emitting this energy in the form of electromagnetic radiation. This would leave the electron with a tiny bit less energy and this it would cause it move a bit slower than before. The centrifugal force keeping it apart from the atomic center would be a bit less, and thus the electron would have to move a tiny bit closer to the center of the atom. Hummmmm... we have a problem.
The more orbits the electron makes, the more energy it loses, the slower it moves and the closer it is pulled to the atomic center. Eventually, therefore, the electron would spiral down and down and finally crash into the center of the atom.
Rutherford's picture of an atom, while a good one and a big step forward in our understanding of atomic structure, had this one - fatal - flaw; electrons could not "orbit" the atomic center, slowly and constantly give up some of their energy, and then spiral down into the center of the atom. A new idea was needed.
That's when Neils Bohr, a Danish physicist started thinking, and by 1913 he had what he thought was the answer.