What is actually in between the gaps of nucleus and electrons? I questioned myself that between any two objects, what is there in between? We all know it's space or air which in turn are made up of atoms. Now since everything is made up of 3 fundamental particles and between the nucleus and the electrons there is so much gap, and I know that due to wave function of electrons their position can be probabilistic, but it is obvious that at one time it can't exist everywhere around the nucleus, so at that time what is there in between the gaps, obviously another atom can't intrude the space in between those particles and so we can't say that "that gap" is filled with atoms. So "in what" do these fundamental particles exist when everything is made up of them including the so-called gap known as space?
 A: The wave function is fundamental. So there is wave function in the gaps.
Trying to push different wave functions into each other is akin to pushing two objects together. The "at one time" way of thinking isn't conducive to gaining a proper understanding of quantum mechanical interactions.
A: 
I know that due to wave function of electrons their position can be probabilistic, but it is obvious that at one time it can't exist everywhere around the nucleus

Actually you are imagining the atoms, and atoms in an ensemble (solid fluid, gas) in the semi-classical model of Bohr, even though you are talking of the wave function.
There are orbitals, not orbits, even for the simple hydrogen atom, and a model of those is needed in order to be able to see the boundary conditions between atoms and how the space is modeled quantum mechanically.
Let us look at the  2s orbital of an excited hydrogen negative ion.

You are wondering what exists in the space between  the nucleus and the electrons, part of  which is seen in the accumulation  here completely empty in plotting the two orbitals.
At the level of quantum field theory (QFT)  all space is covered with the quantum fields of all the particles in the table of elementary particles, including the space in the pictures.
These fields are acted upon by creation and annihilation operators that create and annihilate the electrons in their orbitals. It is a complicated system, not used in the case of bound states, as the simple quantum mechanical models, with correction using QFT are quite adequate to describe the data and predict new data.
