A saving throw—also called a save—represents an attempt to avoid or resist a threat. You normally make a saving throw only when a rule requires you to do so, but you can decide to fail the save without rolling. The result of a save is detailed in the effect that allowed it. If a target is forced to make a save and lacks the ability score used by it, the target automatically fails.
Page 373 of PHB 2024
Bolded section is what I am concerned with. Does this mean that if I am a Barbarian with Strength/Constitution saving throw proficiency, that I automatically fail when needing to perform a Dexterity saving throw (because I do not have the ability score used by it)? Example here would be the Fireball spell cast by an enemy.
The wording is “lacks the ability score used by it”, not “lacks proficiency in the ability score”. So, your barbarian and almost all creatures have all six ability scores (although they may be low values, even 0). A table or hatstand does not have a Dexterity score so would automatically fail a save against Fireball.