How are atoms supported on each other in a material? Suppose we have a ball made up of iron. There are a "lot" of atoms in the ball.
My question is "how" are the atoms supported on top of each other?
And, is it due to the repulsion of electrons the atoms maintain distance between themselves?
 A: The atoms in any object never come into contact with one another for exactly the reason you stated - the outermost electrons repel. Furthermore, the boundaries of an atom are ill-defined in quantum theory, meaning that to discuss contact between them does not make much sense.
In the case of iron, a metal, the positive metallic ions are strongly attracted to free elctrons within the metal. In this way the arrangment is very similar to that of water, with a "surface tension" cause by the continuous electric force between the ions and the electrons. The balance between repulsion and attraction is a very stable one in metals. 
A: Atoms in a solid are held in position by bonds which form between the atoms. Those bonds consist of either 1) electrons which are simultaneously shared by two atoms, yielding a covalent bond, 2) electrostatic forces of attraction which arise when one atom donates an electron to another, yielding an ionic bond, or 3) the electrostatic forces of attraction which arise when all the atoms in the solid share electrons with all the other atoms in that solid, giving rise to a metallic bond. 
The electrons that are involved in these bonding processes are generally in the outermost orbitals surrounding the atoms, which are referred to as the valence electrons. 
