Why is cyclopentyne unstable?
Stable alkynes are normally linear, that is, they have bond bond angles of 180°. That is necessitated by the fact that alkynes are formed by two π-bonds and one σ-bond, which must be mutually perpendicular. If cyclopentyne were to exist, it would be forced to adopt bond angles of roughly 108°, equal to those of the interior angles of a regular pentagon. This deviation from ideal geometry causes extreme instability by virtue of poor orbital overlap. The rigidity of the alkyne moiety would also exacerbate the inherent conformational inflexibility of small rings, wherein the inability of bonds to freely rotate results in eclipsing between atoms.

The only cyclic alkyne I'm aware of is benzyne. However, it's far from stable, existing only as a fleeting intermediate in specific nucleophilic aromatic substitution and pericyclic cycloaddition reactions.

Edit: After a quick search, I found the cycloalkyne Wikipedia page, which claims that cyclooctyne is the smallest cycloalkyne stable enough to isolate and observe. I don't know whether this information is completely current, however.