How can there be current without charge?
From the example I understand that the net charge vanished but that there can be a current. But I was wondering whether this is just a good approximation and that actually one could take into account that there exist charges.
Nobody is saying that charges don't exist. If $\nabla \cdot \mathbf E = 0$, then that means that in every volume of space, there are an equal number of positive and negative charges (which could be zero or non-zero). If you consider a classical model of electrons and protons where they are simply tiny charged spheres, then this is an approximation which would break down on atomic length scales, but this would be very quickly averaged out on larger scales.
This is an interesting question and points to a possible misconception induced by the usual presentation of the subject.
The facts are that, at the macroscopic level, we have two quite independent concepts: charge and current. If one would have only the knowledge of Coulomb's law and its consequences and Ampere's law and its consequences, it would be clear that there are two separate families of phenomena induced by two different sources. However, we know much more and we know that there is a relation, encoded in the continuity equation: the divergence of the current density is equal to the time derivative of the charge density. This is directly connected to the experimental fact that we can charge or discharge conductors by means of currents. Still, it is important to recall that a constant current alone does not generate an electric field.
The reason for our belief about the strict relationship between charges and currents originate from the classical model for a current due to the motion of pointlike charges. We can assign to each charge $q$ at the point $\bf r'$, moving with velocity $\bf v$, a current density $$ {\bf j}({\bf r}) = q {\bf v} \delta ({\bf r}-{\bf r'}). $$ if we have two fluxes of an equal number of oppositely charged particles moving in opposite directions, we have a system globally neutral, but with a non-zero current.
Things become even more interesting (or puzzling) in a quantum-mechanics, where we cannot assign to each particle position and velocity at the same time.
Therefore, from the point of view of QM, it becomes useful to maintain a larger decoupling between the concept of current and charge than in the classical case. Even in the case of a single particle the presence of a current becomes decoupled from the ill-defined concept of motion. For instance, we are sure that the electron in the hydrogen atom in the $1s$ orbital does not stay fixed at one position in space. Still, the $1s$ orbital does not carry any electric current. Instead, a $2p_1$ state carries a well definite electric current experimentally measurable through the resulting (orbital) magnetic moment.
Of course, even at the quantum level, there is no current if there is no charge at all. But classical and quantum examples show that the relationship between current and charge densities is looser than usually thought.