If you’ve done the “balloon inside a heated milk jug” experiment, then likely you’ve also done the trick where you heat the air in a milk jug and put the lid on. It buckles pretty easily as the jug cools.
This is the problem you are running into - after 4 pages of comments - there is no material that comes close to the strength to resist buckling with even a measly fraction of 1 atmosphere / 14.7psi, yet light enough to be balloon material.
Scaling does not work. There’s the “square-cube” law. Double the dimensions of a container. It has then four times the area, eight times the volume. this has implications for any design. Structures need to be re-engineered for different sizes.
Your eggshell design is - minus the negative pressure - essentially what a zeppelin is. It has a rigid shape, and is filled internally with flexible helium bags. The problem with a “suck out air to make the helium bag bigger” design is that the helium bag will only expand enough to equalize the pressure. unless you add more helium to the internal bag, you still have, say, a 40000 gallon shell where you’ve sucked out, say, 100 gallons of air. the size of the helium bag inside makes no difference - the total weight of the system is then 39900 gallons of air minus the equivalent weight of air to match the volume of the helium at 1atm. plus the weight of that helium at 1atm. - plus weight of bags, shell, etc. the problem is that even sucking out 100/40000ths of air creates a pressure differential that can only be held to shape using unobtainium.
A rope or paper does not get weaker as it gets bigger. The problem with the scaling exercise described is that rope or construction paper in extreme sizes must also carry itself. A rope stretched across 10 feet with X strength is holding, say, a 100lb weight - a little over 100lb tugging at both ends’ anchor points. Stretch it across 1000 feet, and each end is holding the same 100 weight but also 1000 feet of rope’s weight which is not trivial.
Not sure if you tried walking on your pool tarp. Probably not fun. The tarp is not resisting compression; it’s resisting tension. First, at the edge of your feet (or the elephant’s) there’s a stretching force. Let’s say the elephant has 4 feet, each supporting 2000 lb (He’s walking, so not always will all 4 feet have even weight. It’s a 1-foot-diameter footpad. That means around the circumference of the elephant’s foot, a distance of 3.14 feet, is supporting a tear (tension) force of 2000lb or 2000/37=54 pounds. In other words, if you hung 54 pounds on a 1-inch-wide strip of this material, would it support it? Probably yes. Not rocket science, reduced to spherical chickens…
OTOH, the pool tarp has no resistance to buckling. If you or Dumbo walk on it, you sink. (That’s what would happen to your zepplin shell - it would simply collapse). you sink in until the dent you make displaces enough water to balance your weight. If Dumbo is too heavy, then it becomes a suspension bridge equation, where the unbalanced weight becomes suspended by the anchor points on the edge of the tarp, and depends on the taughtness of the tarp installation. the pool tarp is a shell simply because the internal material - water - is far more dense than the external material, air and only occasional elephants.
I’d read the fine print. What they are probably saying is that you or Dumbo will not rip through the fabric and sink, but will be able to flounder until you can crawl to safety. This is probably reinforced by the tear-resistance of the tarp, so even wearing high heels you will not create a hole that will continue to grow until you fall through.