If you take your hand and push it against your desk or table you'll be met with resistance. That's the negative charges of the electrons surrounding atoms repelling each other (just like in magnets). It's why you can't phase shift through walls and what makes our universe tangible (physical/solid). If it weren't for electrons all of the atoms in your body would slip between those in your chair and then the floor and you would fall straight through to the center of the Earth. So what keeps everything from flying apart if everything is being repelled by each other's electron fields?
The short answer is that interactions between atoms keep them together. Atoms exchange smaller particles with each other this act of interacting keeps the whole universe from simply evaporating.