That diagram wouldn't allow for electron sharing, though.  If electrons can be shared between atoms to create and build molecules, then the outermost shells of the adjacent atoms must be connected by whatever pattern the electrons themselves follow.  Rather than two discrete spheres, perhaps they would be have to be more like two teardrops connected at the thin ends. 

If those dumbell-like constructions had enough energy that they could tumble over each other, the substance would be a liquid.  Held from tumbling, it would be a solid.

From another article, this one discusses the different bonds.  Covalent bonds are described -

In covalent bonding, electrons are shared between the molecules, to saturate the valency. The simplest example is the H₂ molecule, where the electrons spend more time in between the nuclei than outside, thus producing bonding.

So if the two pointed-teardrops image is correct, the shared electron(s) would spend most of their time within the teardrop points as opposed to circling one atom or the other of the pair.  Once again, this "bridge" indicates a physical connection, and I don't see how the bridge could be substantially longer than the atomic radii themselves.  The bridge gives the pairing enormous strength, so it can't itself be flimsy.

edit fur spellun

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The diagram precisely represents the description you posted.

Important to remember, too, that the familiar model of the atom is just that, a model. Something that gives scientists a way of describing the actions of the various parts -- which are, themselves, only models.

We have no idea what an atom really "looks like".

The radius of a circle is half its diameter. If the radius of an atom is half the distance between adjacent atoms, then that distance would be equal to the diameter of the atoms.

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That section of the page isn't written clearly, but the atomic radius is half the distance between the nuclei of adjacent, identical atoms.

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I think they are referring to the idea that in those states, with just neutrons, the actual nuclei of the atoms are touching.

Collapsed (degenerate) matter has the electrons forced inward until they are touching the nucleus, joining with the protons and forming nothing but a sea of neutrons.  Hence, neutronium.  All the empty space between the nucleus and electron shells is gone, so a given amount of matter takes up a much, much smaller volume.  Staying with your earlier analogy, JB, the empty football field is now totally covered in houseflies!

Pressures at Jupiter's core force the atoms of hydrogen gas together until they shed their electrons, creating metallic hydrogen.  Temperatures are too low for fusion to occur, so Jupiter is sometimes refered to as a protostar. LINK.

There's lots of interesting stuff that happens at a high enough gravity.

edit to add link

Jon, some solids are nothing but atoms.  Carbon crystals (diamonds) are made of carbon atoms and nothing else.  Each carbon atom holds onto 4 other carbon atoms through covalent bonds.  Imagine a a stool with 3 short legs and a post the same length as the legs sticking up out of the seat.  Attach other stools at each post or leg, and continue building.  Nothing but atoms and covalent bonds, and the hardest naturally occuring substance we have on earth. 

Coal and graphite are also pure carbon with different crystal patterns from diamond.  Refined metals like gold and silver are just gold atoms and silver atoms.  They are in molecular form (ore) prior to refining, atomic form (pure metal) after refining.  There are lots of examples of "atoms only" all around you.

(10K gold, or 14K or 18K are molecules, but pure 24K gold is just gold)

Another bit on metallic hydrogen that includes a definition of "metal" that describes iron atoms as being "in contact" with each other.

LINK