For an organic molecule to be completely linear, all the carbon atoms making up the chain must possess a linear geometry. According to VSEPR theory and hybridisation concepts, $sp$ hybridised carbon atoms have a linear geometry with a bond angle of $180^\circ$.

• $\text{HC}\equiv\text{CH}$ (Ethyne): Both carbon atoms are $sp$ hybridised, resulting in a strictly linear molecule.
• $\text{CH}_3-\text{CH}_2-\text{CH}_2-\text{CH}_3$: All carbon atoms are $sp^3$ hybridised, giving them a tetrahedral geometry ($109.5^\circ$ bond angle) and making the carbon chain zig-zag.
• $\text{CH}_3-\text{CH}=\text{CH}-\text{CH}_3$: Contains $sp^3$ (tetrahedral) and $sp^2$ (trigonal planar) hybridised carbon atoms, making the molecule non-linear.
• $\text{CH}_2=\text{CH}-\text{CH}_2-\text{C}\equiv\text{CH}$: Contains $sp^2$, $sp^3$, and $sp$ hybridised carbon atoms, making the overall structure non-linear. Therefore, ethyne is the only linear molecule among the choices.