When I leave for Korea next year, I'd like to have an address to send purchases I make while I'm over there.

Last time I sent them to my cousin but that wound up taking up an entire closet in his house by the time I left. I don't think he wants to do that again.

Is anyone aware of a long term storage address service or anything similar?

I'll be over there for 2-4 years.

Maybe rent a small storage unit near your cousin. He can drop you stuff off in it every so often to keep it from cluttering up his house.