What makes a Horcrux object (almost) indestructible?

Question

We know that a Horcrux is the object in which a piece of the Horcruxee's Soul is stored in.

But these objects were once normal (albeit special) objects before being turned into a Horcrux, and could be destroyed like any other objects.

My question is: what is it about becoming a Horcrux that makes the object (almost) indestructible?

Is it that a powerful enchantment is placed on the object upon Horcruxation? Or something else?

My thinking is that it's not so much the object, but the piece of soul which is difficult to destroy. As Hermione puts it:

"Because a Horcrux is the complete opposite of a human being.”

Seeing that Harry and Ron looked thoroughly confused, Hermione hurried on, “Look, if I picked up a sword right now, Ron, and ran you through with it, I wouldn’t damage your soul at all.”

“Which would be a real comfort to me, I’m sure,” said Ron. Harry laughed.

“It should be, actually! But my point is that whatever happens to your body, your soul will survive, untouched,” said Hermione. “But it’s the other way round with a Horcrux. The fragment of soul inside it depends on its container, its enchanted body, for survival. It can’t exist without it.”

-Harry Potter and the Deathly Hallows - the Ghoul in Pyjamas

So in a way, the object is linked to the piece of Soul's survival.

Answer 1

The Horcrux isn't innately durable on it's own; part of the creation process (not required, but recommended in Secrets of the Darkest Art) is enchantments to protect it. From right before the quote you provide, Hermione explains:

“No,” said Ron, before Harry could answer. “So does it say how to destroy Horcruxes in that book?”

“Yes,” said Hermione, now turning the fragile pages as if examining rotting entrails, “because it warns Dark wizards how strong they have to make the enchantments on them. From all that I’ve read, what Harry did to Riddle’s diary was one of the few really foolproof ways of destroying a Horcrux.”

[...]

“It doesn’t have to be a basilisk fang,” said Hermione patiently. “It has to be something so destructive that the Horcrux can’t repair itself. Basilisk venom only has one antidote, and it’s incredibly rare—”

[...] That’s a problem we’re going to have to solve, though, because ripping, smashing, or crushing a Horcrux won’t do the trick. You’ve got to put it beyond magical repair.”

[Emphasis added]

The fact that it references 'enchantments on them' suggests that the the spell itself doesn't protect the item, but that the wizard must cast spells upon it to do so. More to the point, it seems to suggest sufficient spells to restore the item from all but irreversible destruction.

So, in general, the container is made as tough as possible by other spells; since it's not in a living body (as a general rule) that can repair itself, it's enchanted to self-repair. The situations where we see a living host, it's much easier to destroy, as it's missing (at least MOST of) the protective spells -- a quick 'kill' and the soul is freed (see Naganini & Harry); no need for greater destruction as no magic can restore the dead to life in HP. (Not really surprising that enchantments to restore non-living objects are a bit more comprehensive than those for living items; otherwise the whole Horcrux idea would be unneeded.)

Answer 2

The act of making it a horcrux itself does not affect the object's durability as far as I know. Of course, it's in the dark wizard's best interest to protect it to the best of his/hers ability.

To permanently destroy the horcrux (and the soul part living inside it) it has to be destroyed to the point that it is impossible to repair magically.

This is why only very powerful agents such basilisk poison and fiendfyre have been shown to destroy a horcrux, as only those are powerful enough to render objects unrepairable.

Answer 3

Horcruxes get strength from protective spells, not the soul piece.

A Horcrux isn’t strong because of the piece of soul in it, it’s made to be strong because the wizard creating it would use spells to ensure that it’s difficult to destroy, as they’re encasing a piece of their soul in it. It’s not the piece of soul itself that give Horcruxes their near-indestructible quality. Despite the piece of soul in him, Harry shows an example of something with a piece of soul in it, but without any of the the enchantments that Horcruxes typically have. Harry is still just as vulnerable to injury as any other wizard, as he’s injured several times and isn’t either less vulnerable or quicker to heal than other wizards. He’s also vulnerable to injury by mundane means, like being hit really hard.

“WHAM!

He had stayed still a second too long. The Bludger had hit him at last, smashed into his elbow, and Harry felt his arm break. Dimly, dazed by the searing pain in his arm, he slid sideways on his rain-drenched broom, one knee still crooked over it, his right arm dangling useless at his side.”
_{^{- Harry Potter and the Chamber of Secrets, Chapter 10 (The Rogue Bludger)}}

This definitely wouldn’t work on a proper Horcrux, as they can’t be damaged by smashing - they can only be damaged by powerful substances that put the object beyond magical repair.

“That’s a problem we’re going to have to solve, though, because ripping, smashing or crushing a Horcrux won’t do the trick. You’ve got to put it beyond magical repair.”
_{^{- Harry Potter and the Deathly Hallows, Chapter 6 (The Ghoul in Pyjamas)}}

Horcruxes are nearly indestructible because wizards making them also strengthen the objects they use. The piece of soul dies if the object it’s in is destroyed, so the wizard making one would want to ensure they make it difficult to destroy. The instructions on making Horcruxes warn the wizards who intend to make one that the enchantments on them have to be strong, implying that if it’s not protected strongly enough by the spells being put on it, the Horcrux can be destroyed easily.

“So does it say how to destroy Horcruxes in that book?”

“Yes,’ said Hermione, now turning the fragile pages as if examining rotting entrails, ‘because it warns Dark wizards how strong they have to make the enchantments on them. From all that I’ve read, what Harry did to Riddle’s diary was one of the few really foolproof ways of destroying a Horcrux.”
_{^{- Harry Potter and the Deathly Hallows, Chapter 6 (The Ghoul in Pyjamas)}}

Therefore, it’s clear it’s not the soul piece that makes the object nearly indestructible - otherwise Harry should also be nearly indestructible because of the piece of soul in him, and there’d be no need to further strengthen Horcruxes using protective enchantments.