As a primary source, the Monster Manual (Page 312) explanation for nonabilities (not a score of zero, but lack the ability altogether) is:
Nonabilities: Some creatures lack certain ability scores. These creatures do not have an ability score of 0—they lack the ability altogether. The modifier for a nonability is +0. Other effects of nonabilities are detailed below.
The section detailing constitution specifically states (My emphasis):
Constitution: Any living creature has at least 1 point of Constitution.
A creature with no Constitution has no body (a spectre, for example) or no metabolism (a golem). It is immune to any effect that requires a Fortitude save unless the effect works on objects or is harmless. For example, a zombie is unaffected by any type of
poison but is susceptible to a disintegrate spell. The creature is also immune to ability damage, ability drain, and energy drain, and automatically fails Constitution checks. A creature with no Constitution cannot tire and thus can run indefinitely without tiring (unless the creature’s description says it cannot run).