Shoe storage can get out of hand for us, too (only two kids for now, but we live in about 700 sq ft). It goes along with the fact that I hate shoe shopping and we don't where shoes indoors. My solution has been that I only have two pair of shoes at any given time (loafers or boots in the winter and sandals in the summer along with season-appropriate formal shoes). My casual shoes usually get worn out in a season, and I store the formal shoes for longer term use.
The kids also get two pair of shoes at any given time--casual and formal. Although, right now, ds1 only has one pair of shoes, because I couldn't find a second pair of shoes in his size at the thrift store. I can't afford anything better than Walmart if I buy new, and those always seem to fall apart after a couple months. Shoes that survive the season get stored to pass down, and I'm always on the lookout for the next size up at the thrift store. In any case, only having two pair at a time helps limit how much of a mess is even possible.
Of course, it was really easy for my grandmother's family (during the Depression). Everyone got one pair of boots when the weather cooled. They would be worn out by the time summer hit, and everyone went barefoot for the summer. Can't do that now, and I wouldn't want to, but I thought it was worth mentioning.