When the atoms are very close together, the nearby atomic nuclei are very near the outer electrons and their positive charge can also have a significant effect; the sum of the forces on the outer electron away from any particular nucleus is the same as the force of attraction. In many (most) solid elements, this is the case and this leads to very mobile outer electrons (so much so that they are described as dissociated). The overall result of this is that the substance tends to be relative dense because all the nuclei plus the rest of the electrons (referred to as positive ion cores) are held together closely by this net of free electrons. This effect is called Metallic Bonding and accounts for the common characterises of many substances - which are given the common name of Metals - they are strong and can distort without the bonds 'letting go' (in the pure substance) and they are good conductors of both electricity and heat due to the highly mobile electrons.
NB the non metal solids behave differently because of the particular energy levels of the outer electrons and this can result in Ionic and Covalent bonding. Strangely, it used only to be covalent and ionic bonding that was taught to kids in school but I believe metallic bonding has finally been introduced into their experience. Also, the good heat conduction mechanism in metals was not really acknowledged but used to be 'explained' in terms of vibrations along with heat conduction in non metallic substances.