Since protons and electrons are oppositely charged, what prevents them from attracting to each other and just sticking together?
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You’ve opened up the can of worms that’s called Quantum Mechanics. Simply put, there’s a minimum amount of energy that physics allows in any system, and that minimum is always greater than zero. In an atom, that zero energy would be with the electron and proton being in the same spot. So, instead, it stays just outside the nucleus, hovering just above that zero that it can never reach.
When scientists first discovered that electrons are situated around a dense atomic nucleus, they had the same question as you. It was the quantum equations, ironically derived from some of the conclusions of the Theory of Relativity which is famously incompatible with them, that showed science how this could work.
Protons and electrons do attract each other due to their opposite charges. However, they don’t stick together because of quantum mechanics. Electrons exist in a cloud-like region around the nucleus called an electron shell. They can’t get too close to the protons because of a principle called the Heisenberg Uncertainty Principle, which states that we can’t know both the position and momentum of a particle with absolute certainty. So, while the electrons are attracted to the protons, they also have enough energy to keep them in motion and prevent them from falling into the nucleus.
Congratulations, you’ve just broken classical mechanics. Had you asked this question in 1880, you’d be halfway to winning a Nobel Prize.
Yes, the protons and electrons are attracted to each other. According to classical calculations, the electron of a hydrogen atom would collapse into the nucleus in 10^-33 seconds. Obviously, it doesn’t, so the question is “why?”
It turns out the electron is only allowed to be a certain distance away from the nucleus. It physically can’t be any closer than at its “allowed” orbital, and this distance is more than 0. That might not make sense to you, but quantum physics doesn’t.
An electron and a proton getting stuck together is called a “hydrogen atom”.
As others have mentioned, quantum mechanics puts a limit on how tightly they can get stuck, because there’s a single lowest-energy state, the ground state, or what a chemist would call the 1s state.
The answer to why there’s a lowest-energy state is less ELI5, but: electrons obey the Schrödinger equation. If you plug into the Schrödinger equation the force on the electron due to attraction by a proton and solve it, the solutions are a set of functions called the [spherical harmonics](https://en.wikipedia.org/wiki/Spherical_harmonics), and out of that set of functions there’s one which has the lowest energy, so that’s the ground state.
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