Imagine a world so brilliant and yet unthinkable - a habitable Earth-like planet infested with intelligent life attempting to erect a flexible Dyson sphere enveloping two yellow hypergiants. I know that a hypergiant is rare, and a twin system with a goldilocks zone around it is millions or billions of times rarer still, let alone trying to imprison the hypergiants behind bars to extort its enormous energy output. My question is, can such a world exist outside of my mind?
An out of the box answer would be to use the process of star lifting to get the needed mass for a Dyson swarm or a more ephemeral Dyson cloud (a Dyson cloud uses light pressure to stay in place. To give you an idea of how ephemeral it would be, one around our own sun would mass as much as a medium asteroid). A solid Dyson sphere is impractical for multiple reasons even around our own sun, so I will only consider a Dyson Swarm.
Star lifting involves using a ring of article accelerators around the star to generate a magnetic field to "squeeze" the star. The mass is ejected from the poles of the star where it can be collected. The plasma can be passed through increasingly powerful fusion generators to create heavier elements. The Hydrogen and Helium can also be cooled and used to create gas giant planets for temporary storage until needed.
This allows you to get as much mass as you need, and has the added benefit of reducing the mass of the star and allowing you to control the rate of fusion in the core. A yellow hypergiant is thought to be near the end of its life and the core is already fusing heavier elements. Once it reaches iron, the fusion process produces no net energy and the radiation pressure stops, no longer countering the gravitational pull and the mass of the star fall inwards to trigger a core implosion. Once that happens you have a Type 2 Supernova. For a brief period of time, the star outshines the Galaxy and the massive energy release in the core creates all the heavier elements. Your civilization could be using the hypergiant stars as mines for heavy elements, and star lifting as a means of fine tuning the process.
It seems highly unlikely. finding enough material to put a Dyson sphere around our small sun at a distance we could find habitable would be a challenge, it would use up most if not all of the material orbiting the sun to do it. Now taking a much larger sun, and actually having 2 of them that need to orbit each other in the center of this sphere makes it even harder.
On top of that, the purpose behind a Dyson sphere is to catch energy to be used for the civilization. What kind of use would a civilization need to put the full energy output of two yellow hyper-giants toward? a Dyson ring which would still be difficult to build around a dual system would be much more likely. But I think that just having two suns would make the construction almost impossible, and as other questions have postulated, it is unlikely that you'll find a stable planetary system in a multiple star system.
I would say it is highly unlikely. If you consider the fact that 99% of the mass in the solar system is that of the sun.. Imagine the materials needed to construct the Dyson Sphere. The civilization would have to plunge the materials needed from other solar systems and transport them. It seems a very impossible task.