What is an element?

Or if not annihilating with the electrons that were (very briefly) orbiting them, with other electrons from other atoms in the vicinity, to much the same effect.

I think there is a misunderstanding here, but it may be a linguistic problem: maybe you use the word element differently in English. At least in Spanish and in German I believe the property of being an element is not determined by an atom of that element being in monoatomic form or in a pure molecule (O2, O3, N2…). Poliatomic molecules are composed of elements too: NaCl is composed of an atom of the element sodium and an atom of the element chlorine. Both atoms are still elements despite being in a compound. The right unit to determine whether something is an element is the individual atom, independently of what other atom this atom is bound to, or, like in the case of the noble gases, not at all.
So at least in Spanish and German it would not be correct, strictly speaking, to say that O2 is an element, though when people say that “oxygen is an element” most people will think of molecular oxygen. O2 is not an element but a molecule composed of two atoms of the element oxygen.
If that sounds too much like splitting hairs in English this is OK with me. But that an element can only be defined at the level of the individual atom is the answer I would give to the OP. Where that atom happens to be, in a molecule with only this element or with different elements or in monoatomic form, and whether it is in an electrically neutral form or ionized, can be described independently of the fact that it is this or that element.
Isotopes can also only be distinguished at the level of the individual atom. You may have a mixture of C12 and C14, but the decision whether one atom happens to be a C12 or a C14 atom has to be made measuring the mass of that one atom. The average carbon atom has an atomic weight of 12.011, but no atom does have such a weight in itself, that is a statistical value. It is either 12 or 14 (plus the other isotopes, weighed accordingly).

I always though if an element as being something conceptual. Oxygen is an element, but an O, O2, or O3 molecule isn’t an element.

O2 is composed of two atoms of the element oxygen, therefore it is a molecule of the element oxygen. Put a bunch of those molecules together, and you can have a whole tankful of the element oxygen.

My own feeling is that an element is not a concrete thing at all, but a conceptual class. Carbon is the class of all atoms whose nucleus includes exactly 6 protons. Now what is an atom? Well, the word comes from two Greek roots and means uncuttable. Since fission is possible, atoms can be cut, but that changes their nature. The reasoning is getting a bit circular.

Yes, I think that lines up with what I was trying to say.

It doesn’t matter what the linguistic root of the word is, though. Atoms are a defined thing; a nucleus of one or more protons plus possibly one or more neutrons to which are bound one or more electrons. The fact their nature isn’t consistent with what the Greek word used to mean doesn’t matter.

Atoms matter!

OK, I walked into that one.

And this is why He-3 being stable puzzles me, because it has only one neutron to corral two protons, as it were.

How would that even be atom, with almost no mass? Did you mean “protons” instead of positrons?

When you have only a few nucleons, like the 3 or 4 in helium, each of them is in direct contact with all of the others, so everything happens at very short ranges. At such short ranges, the Strong Force is so much stronger than the Electromagnetic Force that electromagnetism is almost completely irrelevant. The protons don’t just both stick to the neutron which holds them together; they also stick to each other.

If it were just up to the Strong Force, nuclei would be most stable with an equal numbers of protons and neutrons, and indeed, that’s what happens with the smaller elements. But once you get up to the very large elements, the average distances between nucleons is larger, and so the longer-range electromagnetic force becomes more relevant compared to the shorter-range strong force, and so too many protons starts to be a problem. And so the heavier elements have more neutrons than protons.

Hm, if that’s the case, why isn’t 2He stable? With only two protons, they’d both be in direct contact with each other.

Wikipedia writes it is due to spin-spin interactions in the nuclear force, and the Pauli exclusion principle.
(Ha! Second time today I mention Pauli’s exclusion principle! That hasn’t happened in a long time!)

Sodium chloride is an interesting example when talking about molecules. Think about the sodium and chloride atoms in a solid salt crystal. Each chlorine atom is equally adjacent to six different sodium atoms and each sodium atom is equally adjacent to six different chlorine atoms. It would be hard to say which chlorine atom and which sodium atom should be paired up to make a single NaCl molecule. It might make more sense to think of the entire crystal as the molecule. Similarly think about a solution of sodium chloride in water. Sodium chloride is completely dissociated in water. The individual sodium and chloride ions “floating” around in the water have no idea of which of the other gazillion ions in the solution they were next to before they got dissolved. However, sodium chloride molecules consisting of a single sodium atom and a single chlorine atom might be present in the gaseous state (along with the dimer disodium dichloride and, who knows, maybe the trimer trisodium trichloride) when solid sodium chloride is melted and then boiled. But they are probably rarely encountered in significant quantities by most people on this planet because sodium chloride boils at a couple of thousand and some change degrees Fahrenheit.

The Strong Force can either attract or repel, depending on a complicated and not-fully-understood set of circumstances. Suffice it to say, though, that it treats protons and neutrons as distinct but not really different, sort of like the colors of cards in Solitaire: There are rules that say that you have to alternate the colors in the stacks, but the rules treat red cards the same way that they treat black cards.

Under the rules of the strong force, two of the same kind of baryon aren’t stable, but one of each, or two of one and one of the other, are stable*, and two of each (a helium-4 nucleus or alpha particle) are most stable of all.

*Yes, tritium (two neutrons and one proton) is unstable, but that’s due to the Weak Interaction, not the Strong. When something’s unstable due to the Strong Interaction, it typically flies apart or otherwise decays in a trillionth of a trillionth of a second, or so, while tritium typically lasts for decades.

There is mass, though (and greater mass tends to increase decay rates, as a rule of thumb). For regular positronium (1 electron + 1 positron) the mass is 1.022 MeV and the binding energy is a few eV— not extremely stable but definitely a bound state.

What about most gaseous elements that aren’t monatomic? If you have one oxygen atom hooked up to one or two additional oxygen atoms, you can’t call that an “atom”, so it has to be called a molecule, doesn’t it?

There’s a difference between saying an element has molecules, and saying that an element is a molecule.

I’ve been following this discussion with interest. I’ve tried to combine the material posted here into some working definitions. I hope someone can offer some improvements.
atoms: (an element/can be an element??) consists of a nucleus of one or more protons plus possibly one or more neutrons to which are bound one or more electrons (One can quibble about the number of electrons in an atom)
an atom can be an element ( single atom** of Californium is…an element…** So is two atoms of californium, or ten atoms, or 10^24 atoms of it.)
atoms can be elements despite being in a compound

elements: cannot be chemically interconverted or broken down by any chemical reaction into simpler substances and are primary constituents of matter. Each element is distinguished by its atomic number, i.e. the number of protons in the nuclei of its atoms. Elements have a fixed number of protons, and in a neutral state have the same number of electrons.
element: a substance which consists entirely of atoms all of which have the same number of protons. a single atom of Californium is ..an element… So is two atoms of californium, or ten atoms, or 10^24 atoms of it.

molecules: (formed by 2 or more atoms) is never an element.