Dungeons and Dragons

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.

No, it would mean if a target didn't have any dexterity that they would automatically fail it. I believe this would mainly be applying towards objects/the environment, but there are probably other contexts as well.

A lot of people are noting ability score of 0, but I think the wording was meant to be applied to more inanimate things. An ability score of 0 is just a -5 to the score. Lots of sentient magic items and objects will have limited stats. For instance, Whelm is a magic sentient warhammer with an Intelligence of 15, Wisdom of 12, and Charisma of 15. If, for instance, it was forced to make a Strength saving throw, it has no Strength score, and therefore fails. Likewise vehicles often have Strength, Dexterity, and Constitution scores, but no mental scores. If you forced it to make a mental save, they would automatically fail.

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)?

Unless your barbarian has a 0 in dex, you have a dexterity ability score. Your barbarian has proficiency in Strength and Constitution savings throw, meaning you add your proficiency bonus to your ability score bonus. The other ability scores have a savings throw of the ability score bonus, and any other bonuses you may have, but not your proficiency bonus.

By my reading, your hypothetical Barbarian does have a dexterity ability score whether or not they have proficiency in the saving throw. If they wanted to say lacks proficiency they would have said that directly. I'm not super familiar with the 2024 rules, so I don't know under what circumstances a target might not have the ability score; maybe immobile objects don't have dexterity?

I can't speak to this ruleset, but for other rule sets I've interpreted that kind of thing as if the target has an ability score of 0, regardless of proficiencies or lack thereof. An unconscious person or an inanimate object, for example.

I'd presume you have a Dex value, just not a very good one.