I store mine in bags, on end, in my craft room. Each bag is labeled for what it contains - delta kites, dual line stunt kites, single line kites, vintage kites, art kites, kites I’ve built, low/no wind kites.
My biggest concern for storing is really for the seasons I don't fly. Either due to inclement weather or no wind. Concerns being really condensation possibilities building up on rolled up kites and laying on each other. I keep them in the closet in my room and I noticed last winter keeping my closet door shut with the heater on, it seemed kinda cold when I would go in after work to grab something. So I began keeping the door open instead to let some warm air in.
Yes, I actually DO hang a few kites from the ceiling(chuckle). They make great conversation starters! I switch them out all the time. 👍
Short term? Agreed with Zuul, in a bag in my vehicle, ready to take out onto the field.
Long term, would be a double-check that all elastics are de-tensioned, then laying down somewhere I know they'll stay dry, cool, and otherwise protected. But really, that's just a sad situation since the kite was meant to be flown.
Retired, mounted on a wall in a place of honor for all to see.