Invalid structure of atoms on a classical scale and wrong assumption for the time of publication (Actual question at the very bottom)
In his 1911 paper "The scattering of α and β particles by matter and the structure of the atom" Rutherford starts with this assumption:

Consider an atom which contains a charge +-Ne at its centre surrounded by
a sphere of electrification coutaining a charge -+Ne supposed
uniformly distributed throughout a sphere of radius R.

He then derives the following:

In order to from some idea of the forces required to deflect an α particle through a large angle, consider an atom containing a positive charge Ne at its centre, and surrounded by a distribution of negative electricity Ne uniformly distributed within a sphere of radius R. The electric force X and the potential V at a distance r from the centre of an atom for a point inside the atom, are given by

$X = N e (\frac{1}{r^2} - \frac{r}{R^3})$
$V=Ne(\frac{1}{r}−\frac{3}{2R}+\frac{r^2}{2R^3})$
...
For the time of 1911, this was an invalid assumption. Also, if the assumption is wrong, you can conclude anything.
There wasn't any quantum mechanics around and one can clearly show that a centre charge closely surrounded by the opposite is an impossible structure for it violates Gauss's law.
To picture this, recall that atoms are neutral. Therefore, for any sphere outside the atom we have:
$\oint_S E \,dA = 0$
Therefore, the field lines of the inner charge must surface, which is impossible if the outer charge is continuous.
Since the distance between the outer charge is also a lot closer, the surfacing inner field would be condensed to almost infinite field strength. The positive field lines also restrict any circulation and pose to many constraints for any wave equation to be solvable, since field lines are known to be continuous.
This image of a stylized (non proportional) neon atom created with an EM simulator using the Runge-Kutta algorithm Github: Electric Field Simulator  shows the behavior:

Note, that for this image, Gauss's law holds true. A full 2D circle around this structure would lead to a summation of zero. The near field of this structure, however, extends too far out for atoms to be neutral.
The results can also be obtained by separating the charges next to each other, and then using a linear transformation to map one side into the other.
Such a structure cannot exist under classical mechanics.
However, it wouldn't be the first time, valid results have been arrived via wrong assumptions. But it is rather strange that this escaped the peer review.
Now the question:
Since atoms are not classical mechanics, but quantum mechanics governed, what trait of quantum mechanics turns the clearly invalid classical mechanics structure into a valid quantum one? What makes atoms neutral again and solves the problem of the field lines having to surface and still being continuous?
 A: Lets make it simple  and look at the hydrogen atom:

which is assumed when drawing the  electric field lines in the image in your question, fixed locations for the charges.
Quantum mechanics does not have orbits, but orbitals for a given time t, i.e probability of measuring the electron and the nucleus (x,y,z). This is the measured locations of the electron :

If you want to image the classical field lines  they would be for that instant

The mathematics of quantum mechanics integrates over the space according to the wave-function, at large distances the positive and negative attraction neutralizes. That is why quantum mechanics is successful in fitting the data.
If you assume that the charges in the image in your question are an instantaneous drawing of orbital points then the same argument holds for neutrality.
For large atoms (and higher energy levels of the hydrogen atom) the orbitals can be such that in some directions the positive nucleus field dominates and in others the negative electrons, which allows for bonding of atoms into molecules and lattices, various working models of chemical bonding exist, afaik.
