For "persistent structure" read "bound structure by quantum mechanical potential sollutions".
At the atomic and molecular level structures arise because the number of atoms can settle at a lower energy level, than when free. This means there is a binding energy that has to be payed for the atoms to be freed from the structure.
the atoms might be moving fast but can't we make the molecules come arbitrarily close so that the force of attraction can hold them together?
In the quantum mechanical world, the word "close" has little meaning. Molecules scattering off each other do have a quantum mechanical probability to fall into a lower energy level by giving off their extra energy as a photon. It so happens that this probability is small for energetic molecules , the smaller the higher the difference in the bound energy level and the kinetic energy of the free molecule. Please note that the energy levels between molecules that keep liquids as liquid are very small.
This is a basic explanation , here you can see the complexity in a particular study.