In the literal backyard of my house? No. Of course, I also support windmills, water reservoirs, lithium mining, and electric car factories and don't want any of those in my literal backyard either. The idea that it is hypocritical to advocate anything you wouldn’t want on your own residential property seems like such an obviously ridiculous idea, I'm not sure it makes sense to rebut. There are countless things society wants (factories, mines, landfills, warehouses, etc…) that don’t make sense to do in residential areas.
In my figurative backyard, like the state I live in? Sure, I would definitely support California creating a nuclear waste storage facility.
This isn't really anything groundbreaking. We already have millions of acres of landfills in the US. Like radioactive waste, trash is basically permanent and will be decomposing for at least hundreds of years. And no one wants a landfill "in their backyard," but somehow we still managed to create the thousands of landfills we have today. We can spare a few dozen acres to build a nuclear waste facility.
Ok, so storage is a problem.
The only problems I would have with it being in my hypothetical back yard would be other people thinking it was dangerous, leading to an urban decay cycle (this also applies to property with unlucky numbers, like 13 in the anglosphere), and/or lots of protesters and/or would-be terrorists not realising how deep the stuff is buried nor now hard it is to shift a multi-ton concrete-filled steel barrel.
From what I know of deep geological storage, I wouldn't mind living in a house on top of one of these sites. Since they're usually in the middle of nowhere, it would probably be even safer than living in a polluted city.
How convenient
What's your point??
How about your backyard?