Me too! One of the most annoying first world problems for sure... I still have lots of stuff, mostly old toys that I'm too fond of to just throw away or to take to a recycling shop, at my parents' place. They don't nag at me about it (big old farm house with lots of attic space), but I feel like I should sort the stuff out and at least pack it all better before everything's ruined. Yes, ruined; I checked things a little bit this weekend and noticed that, since my barbie dolls weren't stored properly, there's a lot of trash and dirt from the ceiling on them. And then there are problems with moisture etc. Gotta check what I can still save at a better time. Fortunately though, my old Bratz dolls were tightly packed and only had some dust on them. They're fine; I brought them home and sorted them out. I brought my legos too, though I've yet to go through those and didn't find the two big boards that I have that can be built on.
There's no other dimension to store stuff in as far as I know, but it is possible to rent extra space just for storage. Assuming one can afford it, of course. Could you?
Another option would be to sell stuff at flea market. I mean, at least to me, it's easier to give up on my stuff when I get money for it.
And then there are creative and convenient storage solutions, which could probably be found by googling those words... If I ever need to get a new bed, I want to get one with a box installed at the bottom. I could keep some boxes under my current bed, but that'd make cleaning harder so I haven't done it so far.