Does there/or could there exist a molecule made only of carbon, that is smaller than C20?
Yes. C2 is a molecule Smaller than C2O. Whether it's a naturally occuring one, I'm not sure.
As far as I knew the bonding for C2 is not possible, that's why I asked. Maybe I was misinformed. Perhaps its just that it does not occur in nature. How would the bonding work?
I think C-C will wander until it picks up some -H for example, to make ethane.
C2 exist but it is just very unstable, it's still a molecule.
Well if that is allowed, geometrically I can imagine C4, formally a square with two acetylenic bonds, or a tetrahedron with two double bonds. Both have very strained bond angles but would they get some resonance stabilisation? Could it be made difficult for them to decay into anything if e.g. in dilute gas*?
*for the reason below.
C2 is certainly very stable to dissociation and has been characterized completely in the gas phase. However it cannot be filled in bottles as there are other carbon modifications which are even more stable and to which C2 quickly reacts.
Neither, C4 (if it exists) is planar and rhombic (since the double-bonds are shorter). I'm doubtful it's been synthesized, and I'm doubtful of its stability. There are some recent theoretical studies of its radical ions, though.
There's tetrahedral N4, which I'm a big fan of. Never synthesized either, but I know that one's supposed to be somewhat stable (energetic barrier to decomposition of around 10 kcal/mol or something). It's a very high-energy compound (something like 50 MJ/kg IIRC). It's been proposed as an environmentally-friendly rocket fuel. If someone could come up with some way of making it.
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