What is the net charge held in a battery?
It is correct that the total charge held in a battery is 0, as it is for a capacitor.

(Dis)charging a capacitor means moving charge from the one side to the other, while the total charge remains 0.

However, a battery is a bit different from a capacitor, e.g. the voltage is more or less constant over the full lifetime.

The Daniel-cell is one of the simplest batteries. You need a piece of copper, a piece of zinc, a solution of sulphuric acid and a solution of copper sulphate. All four ingredients are neutral, and since charge is conserved (i.e. you can not 'create' positive charge without getting the same amount of negative charge), the total charge will still be zero, when you mix them together.

I don't want to write down the chemical reactions inside the cell. But in general, the zinc dissolves by sending positive metal ions into the liquid while electrons are left back in the metal. On the other side, copper ions from the liquid attach to the piece of copper and absorb electrons.

In principle, the one reaction is the reverse of the other, but since zinc is less noble, it has 'more power'1 and overcomes the copper. 

Finally, this reactions only takes place when there is an electrical connection between the two pieces of metal, i.e. when a current can flow.
This means that in contrast to a capacitor, a full battery does not contain all its charge yet, rather it is generated during usage.



1 This 'force' - or more the difference in forces of both metals is the voltage measured at the terminals of the battery. The materials define the voltage, and this is why most batteries have about 1.5V, Lithium cells 3V, lead-acid cells (cars!) 2V and so on.