Why is this structure impossible?
Your analysis was a good start, but then it dropped. You need to consider four electrons on each carbon atom. Each of the two carbons can then form a single bond to one hydrogen atom using one of its electrons. Each of the carbon atoms then uses two of the remaining electrons to form the double bond.

Thus, each of the carbon atoms has one extraneous electrons; the structure would be a diradical structure. It should better be represented e.g. like this: $\ce{H-C^.=C^.-H}$.

It is possible for these two radicals to be alongside each other — see for example the structurally strongly related dioxygen molecule which is a triplet (diradical) in ground state and not adequately described in simple Lewis structure formalisms. However, it is highly unlikely for two carbons without additional constraints to form a diradical like that; forming a triple bond ($\ce{H-C#C-H}$) would be much better energetically (for carbon!).