Why own extra tires? Tires are expensive, unlikely to fail unexpectedly, and can be purchased quickly. When a tire wears, I keep the most recent worn tire as an emergency spare. It will easily last the day or two it takes to buy a new tire. In practice, I haven't yet actually used the emergency spare. I ride over 5,000 miles a year, so the emergency spare is never much more than a year old. I'm looking forward to hearing why others think it's important to keep all these tires around for so long that rubber degradation is an issue.
I'm also cutting back on spare tubes. Right now, I have two spare tubes, one in the bike bag and one in the garage. But I think I'm going to stop keeping the second spare. High quality tires and tubes are available. Getting a flat is becoming a rare occurrence. I generally install a new tube when I install a new tire, because the additional cost of a new tube is so low. And by using high-quality tubes, the likelihood of a bad seam or similar failure of a new tube is very small. So the old tube can become my second spare, or emergency tube.
To answer the original question, my emergency spare tire hangs on a hook in my garage. When I go on tour, it's easy enough to pack. I plan to keep the emergency spare tube lightly inflated in the emergency spare tire, hanging off that hook in the garage, taking zero additional space.
When I use the spare tube, it's time to buy a new tube. I sprinkle my new tubes with talcum powder so the rubber is less likely to stick to itself, and to make it easier to install the tube when it is needed. Then I double-bag them in kitchen zip-lock bags. The double-bagging is mostly to ensure the powder doesn't get all over the tools in the under-the-saddle bag.