There are opposing forces.
The electromagnetic force you allude to holds negatively charged electrons in their orbit around the nucleus, which contains one or more positively charged protons.
The nucleus itself, however — protons and neutrons — would fly apart because of the electromagnetic force since their charges are all positive or neutral. However, they do not fall apart because a fundamental force called the **nuclear force** holds them together. The nuclear force is probably four times as powerful as the electromagnetic force so it wins.
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