First off, I'll tell you that I'm a hoarder, in every game I play!
Second, it probably only feels messy and somewhat useless when they're unorganised. So, I'd recommend finding an area on your farm that you think will be the most convenient: your House, a Shed, next to your Greenhouse, or anywhere else, and make it your dedicated storage area.
Organise it in whatever way makes most sense to you, colour code the chests, make sure they're all easy to access, add lighting so you see them at night.
Sort them by season, in-game category, location found, colour, or value, and make sure there's extra space in each chest for items you find later.
Leave 2-4 empty for expansion, and have additional chests to put any items that don't fit in any of your categories.
Then pick a number, 5, 50, 100, or whatever seems most logical to you, keep that many of each item, and sell the rest.
Now you can be proud of the items that you possess, knowing where each and every one is located, confident in your ability to list items that you own.
It probably won't feel like a mess anymore, and you'll be better informed as to what you are throwing away.
Hope this helps!
P.S. That got a bit poetic – I'm sorry! :D