The Hylian Shield isn't technically unbreakable, but it does have a very high durability of 800.
If you haven't mastered the Perfect Guard technique, I'd steer clear of using it against Guardians' laser beams, as they can destroy almost any shield with one blast. While the Hylian Shield might take a bit longer to get worn down by them, it's not worth it when you can sacrifice a stack of low level shields in its place.
The shield is in Hyrule Castle's lockup, but it's very easy to reach. You can avoid all confrontations along the way, but stock up on arrows and various health and speed potions for the unavoidable fight at the end to get it.
From the northwest side of the castle, where the green pin is, you'll be able to see the entrance to the docks, which looks like a big cave with the moat running into it.
Paraglide over onto the path next to it, but be wary of the active, but stationary, Guardian at the top.
Sneak past it to what looks like a mine entrance on the right. There are three ore deposits twinkling away right outside, so you can't miss it.
A gate will be barring your way, but you can use an ice block to raise it up.
The next obstacle is purple Ganon gloop. Shoot the eyeball visible in the cell on the left to clear out the hallway.
The cells here have some tough Moblins in, but you can easily avoid them if you have enough stealth and stamina, climbing above and around the cells like a yellow belly.
Follow the corridor around to the right and you'll see a very suspicious-looking pile of bones in the cell beyond.
It's a Stalnox, and yes, the door will close behind you, trapping you in with it. Shoot its eye and follow up with a sound thrashing. He'll try to throw his various appendages at you, but they're easily avoided. Take your time, keep your cool, and you it'll be dead before you know it.
He'll drop some pretty fancy gear, but the treasure you're looking for is in the chest.
Any armour, weapons, and shields you've acquired, and lost or broken will appear in the inventory of Grante in Tarrey Town, but they're very expensive and you'll need to do a bunch of quests to unlock his services.
So when your Hylian Shield finally bites the dust, you can pick up another one from Grante.