Spiralling of electrons into the nucleus
As a matter of fact, Bohr literally went past around this issue. In 1913, he proposed his model postulating electrons could only have certain classical motions:


Electrons in atoms orbit the nucleus.
The electrons can only orbit stably, without radiating, in certain orbits (called by Bohr the "stationary orbits") at a certain discrete set of distances from the nucleus. These orbits are associated with definite energies and are also called energy shells or energy levels. In these orbits, the electron's acceleration does not result in radiation and energy loss as required by classical electromagnetic theory (This was based upon Planck's quantum theory of radiation).
Electrons can only gain and lose energy by jumping from one allowed orbit to another, absorbing or emitting electromagnetic radiation with a frequency ν determined by the energy difference of the levels according to the Planck relation:
$$ \Delta {E}=E_{2}-E_{1}=h\nu \ $$


where h is Planck's constant.

As remarked above, these are postulates (statements assumed to be true from which further reasoning follows), and (even though that part was left unexplained), it's acceptance was due to the fact it explained correctly many experimental results. It was further displaced by the modern quantum theory.

Hope it helps, it also puzzled me a few years ago! :P

Reference: https://en.wikipedia.org/wiki/Bohr_model