C.,
You didn't mention if your house has a basement or an attic or what types of closets. I live in a 1925 Chic & our bedrooms are all about 9' x 9'. My son has a room, my husband & I have a room, and our third bedroom is a home office plus my husband's dressers & my dresser are in that room since we can not fit any furniture in our bedroom besides a queen size bed & end tables! We don't even have a linen closet or a coat closet, so I have been very creative with storage. For example, our bed is a steel frame platform bed that has about 15 inches clearance below. Plastic drawer type bins under the foot of the bed hold the household linens. Cube shelves in my pantry hold the kitchen towels. Baskets on top of the microwave oven hold dishtowels and pot holders.
My son has a very tall armoire in his room & his closet has been fitted with Elfa shelving (less expensive Rubbermaid or Closetmaid stuff would not work b/c 1925 closets are nowhere near a standard size by today's standards) where I store some toys in bins. In our living room, I have an espresso colored 8 cube shelf by Closetmaid (bought on sale at Target.) All the toys in our living room live in those bins. There are a couple of slightly larger toys in the corner of our dining room. It's not always pretty during the day, but it cleans up fairly neatly. For more room, and to keep our son out of the buffet drawers & cabinets, we push the dining room table over.
Our basement is not finished or heated, but it seems to be maintaining 68 degrees or so. I am clearing out space for when we are desperate if it is rainy or bad weather or just so my son can be down there while I do laundry. Even if we don't spend time playing down there, I am planning to keep a number of his toys on a shelf down there. We can then rotate the out upstairs as the more stuff we have out, the crazier it seems to get. I don't mind that his stuff is all over sometimes, but when I am stepping on or tripping over stuff, it really becomes a problem! Also, I find that the more he has out & pulls out, the more chaotic it is. He is still pretty small, so I would like him to be able to keep the chaos to a minimum so it's easy for him (us) to find toys to play with & then also easier to teach him to clean up his toys. We also have an enclosed, but unheated sun porch that I used in the summer for a play area. It's already too cold, but before I cleaned out the basement & made it as safe as I could, I was thinking that we could always keep larger toys out there & bring them in to play with & return them there when we were done. I suppose if you can not find any storage space at home & can't get rid of stuff, maybe there's a relative nearby (like grandma?) who would allow you to keep toys at their house & trade them out once in awhile. If you do get rid of stuff, you can always check out "toy libraries." Our public library has a toy closet with toys that can be checked out. I hear there are actual independent toy libraries as well.
As for stuffed animals....I hope that people stop giving them to my son. We are overrun with them & he's only 18 mos. old!!!! Most of them "live" up high on the armoire in his room & are more like decoration.
Good luck!