You can craft them together (pre-1.14 or 1.14.3) or put them in a grindstone (1.14+) to make unenchanted items with more durability (the sum of both items plus 5% of the maximum):
Grindstones also accept a single item and give you back an item stripped of its enchantments and prior work penalty, which can then be enchanted again (no more making multiple items or enchanting a lot of books to get the perfect enchantments; both tools/armor and books can be disenchanted so you can simply enchant the same item over and over. You do still need to combine multiple enchanted items to get things like Sharpness V or desired enchantment combinations):
Also, while not really applicable to current versions I save the pickaxes I wear down while branch-mining so I can combine and enchant them again (example, 4 pickaxes, combined in pairs, gives you 3 more opportunities to enchant without making new ones. As mentioned above, you can now use a grindstone on a single item to disenchant it).
Otherwise, you want to get Mending on your items, then you don't even need to worry about worn-out items or having to make new ones (I actually made a mod that reverted the changes to anvil repairing in 1.8, where items could no longer be used indefinitely if you renamed them, because it was so game-breaking to me (even an Unbreaking III diamond pickaxe has a life of only a few days (repaired to "too expensive") given how much I use it), even as I never used it, much less played in 1.8). If they are just items you made early on I wouldn't bother saving them (I have two "types" of gear in my worlds, that which I use during the "early-game" and the "caving gear" that I use during the "end-game").
In the rare case that I wear out an enchanted but non-mending tool or weapon to low durability and it's too expensive to repair on an anvil, I hang it on an item frame on a wall of honor as a "retired" implement. Mostly this tends to be Infinity bows and Fortune III picks, since those are the ones I can't or don't usually add Mending to. Alongside them on the wall of honor goes one non-enchanted item, my first wooden pick from each world I play on.
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"I think I'm starting to like this `programming' thing. It's about four times as fun as shaving." -- Notch, June 12, 2011
You can smelt iron and gold tools for nuggets, but you can't do that with diamond
Is there anything useful I can do with them without simply throwing it out or letting it break?
Assuming you are in the latest version (1.14.3), if the tool is not enchanted, you can use a grindstone or a crafting table to join two tools of the same type and recover some duration. If the tool is enchanted and you don't want to repair it and pay the experience and material cost, you can disenchant it in the grindstone for experience.
You can smelt iron and gold tools for nuggets, but you can't do that with diamond
Is there anything useful I can do with them without simply throwing it out or letting it break?
Edit: I got 2 boots with mending that I currently use, and a chest filled with unenchanted diamond boots.
https://www.youtube.com/watch?v=E5_xb522jEo
You can craft them together (pre-1.14 or 1.14.3) or put them in a grindstone (1.14+) to make unenchanted items with more durability (the sum of both items plus 5% of the maximum):
https://minecraft.gamepedia.com/Item_repair
Grindstones also accept a single item and give you back an item stripped of its enchantments and prior work penalty, which can then be enchanted again (no more making multiple items or enchanting a lot of books to get the perfect enchantments; both tools/armor and books can be disenchanted so you can simply enchant the same item over and over. You do still need to combine multiple enchanted items to get things like Sharpness V or desired enchantment combinations):
https://minecraft.gamepedia.com/Grindstone#Usage
Also, while not really applicable to current versions I save the pickaxes I wear down while branch-mining so I can combine and enchant them again (example, 4 pickaxes, combined in pairs, gives you 3 more opportunities to enchant without making new ones. As mentioned above, you can now use a grindstone on a single item to disenchant it).
Otherwise, you want to get Mending on your items, then you don't even need to worry about worn-out items or having to make new ones (I actually made a mod that reverted the changes to anvil repairing in 1.8, where items could no longer be used indefinitely if you renamed them, because it was so game-breaking to me (even an Unbreaking III diamond pickaxe has a life of only a few days (repaired to "too expensive") given how much I use it), even as I never used it, much less played in 1.8). If they are just items you made early on I wouldn't bother saving them (I have two "types" of gear in my worlds, that which I use during the "early-game" and the "caving gear" that I use during the "end-game").
TheMasterCaver's First World - possibly the most caved-out world in Minecraft history - includes world download.
TheMasterCaver's World - my own version of Minecraft largely based on my views of how the game should have evolved since 1.6.4.
Why do I still play in 1.6.4?
In the rare case that I wear out an enchanted but non-mending tool or weapon to low durability and it's too expensive to repair on an anvil, I hang it on an item frame on a wall of honor as a "retired" implement. Mostly this tends to be Infinity bows and Fortune III picks, since those are the ones I can't or don't usually add Mending to. Alongside them on the wall of honor goes one non-enchanted item, my first wooden pick from each world I play on.
Assuming you are in the latest version (1.14.3), if the tool is not enchanted, you can use a grindstone or a crafting table to join two tools of the same type and recover some duration. If the tool is enchanted and you don't want to repair it and pay the experience and material cost, you can disenchant it in the grindstone for experience.
Name your equipment after what you did with them. Hang them in a museum.
That sword you used to kill your first ender dragon would be called "Wingthief, Slayer of Dragons."
That pickaxe you used to hollow out a mountain would be called "Erosion, Mountain-Eater"