I was at a bus stop today and sat on a discarded inkjet printer. It occurred to me that it would be nice to have brought a stool to sit on while awaiting the bus, but stools are typically heavy and bulky; I'd want something that would fit into my purse (about a liter) without weighing much (the purse weighs about 1kg).
Generally stiff things that can resist compression are kind of heavy, but materials can resist a lot of tension even while being quite light. A simple rope of webbing, like a seatbelt, that could hook onto the roof of the bus stop in two places would make a practical, relatively comfortable, and quite strong hanging seat.
Another way of avoiding the need to carry around materials that can resist compression is to use air to support the compression, as in a balloon. The balloon skin only needs to resist tension, so you can make a pretty large balloon that doesn't weigh much. This couch, almost two meters long, weighs under 3kg: http://www.amazon.com/Blow-Inflatable-Furniture-Sized-Couch/dp/B00245MIAY. A comparably-sized couch made from foam might weigh 30kg.
But how little material could you get away with? I weigh about 100kg (980N) and sit on an area that's about 40cm×40cm, or 0.16m², so an air pressure of 6100 N/m² (6.1 kPa) is sufficient to hold me up. Ambient air pressure is about 101 kPa, so you only need to compress air very slightly to get it up to 101+6 = 107 kPa, say, by sitting on it.
How much material would you need in the skin of the inflatable stool to resist that air pressure? Very little. You can estimate hoop stress for a thin-walled cylinder as Pr/t, where t is the thickness. So the hoop stress of a 25-cm-radius cylinder containing 6 kPa(g) will be 6 kN/m² · 0.25 m / t, or 1.5 kN/m / t.
The biaxially-oriented polyethylene terephthalate film used for mylar balloons and potato-chip bags has a tensile strength of about 200 MPa, almost as strong as low-end aluminum alloys and a lot lighter and more flexible. It's commonly available in 10μm to 100μm thicknesses. To find the thickness needed to contain this pressure, we solve
200 MPa = 200 MN / m² = 1.5 kN/m / t
t · 200 MN / m = 1.5kN
t = 1.5 kN / (200 MN / m) = (1.5 / 200 000) m = 7.5 μm
If we figure on using a film of several times that thickness, say 20 μm, we should be pretty safe from bursting if we don't totally jump on the thing. How much would that weigh? 2πr² + 2πrh, figuring 50cm height and 25cm radius, gives us a total area of 1.2m²; so 20 μm would be 24 cc. (7.5 μm would be 9 cc.) PET weighs about 1.38g/cc, so this would be about 33 g of plastic. This is acceptably small.
That's fine as far as it goes, but trying to sit on a paper-thin mylar balloon at bus stops will have some serious practical problems. In the summer, if you're wearing shorts, it will stick to your thighs in an unpleasant way; but also, the part of it pressing against the rough cement with 6kPa and sliding around as you move will get cut up.
You can solve both of these problems by gluing woven cloth disks to the top and bottom of the plastic, probably the coarser the better. Coarser cloth will weigh more but allow better air circulation and protect more fully against abrasion; as the cloth gets sufficiently coarse, you might want an additional somewhat finer cloth layer between your skin and the burlap or whatever, or between the bottom of the balloon and the ground burlap. This introduces an undesirable tradeoff between weight and performance. Adopting a pattern imitating the reticulate venation of dicotyledon leaves, with a few thick veins separating areas of finer venation, could ease this tradeoff.
The cloth is likely to increase the weight to perhaps as much as 100 g, but we aren't done yet. Now we face the question of how to inflate the thing. It holds about 0.1 m³ (100 liters) of air, which is around 30 lungfuls; inflating it by mouth would be a lot of work to have something to sit on for only, one hopes, a few minutes.
You could carry around a gas cylinder to inflate it with, but this is impractical; 100 liters of CO₂ is about 200 g. You'd need to empty three 88 g paintball-gun CO₂ cartridges to inflate it fully, and those cartridges weigh over 400 g each, so you'd be carrying around 1200 g of cartridges. Even the 200 g of CO₂ itself is unreasonably heavy.
The most promising solution I've seen so far for inflating things like this is the Windcatcher, which uses a sort of funnel with a low-pressure one-way valve to help you entrain air as you blow into the entrance. I don't know how much it weighs, but they were selling a bag using it for US$40 to their Kickstarter backers; a larger air mattress using it weighs 840g, but I hope that's mostly the bag.