We've solved this problem before with libraries. They're one of the institutions that in society are meant to last long-term, and take on the costs of preserving material for perhaps perpetuity. For books, that means a building, shelves, temperature and humidity control, and a staff to watch over all of it - plus whatever above and beyond you want to provide to help people access the material. That's not free but how to pay for it and how to manage acquisitions are well-established.
There are two somewhat separate issues - maintaining a presence online, and deciding where in the namespace it should go.
Certainly it's easy to understand how a library could manage to keep a site online, especially if the site is simply and static. In time (over the next 20 years sort of timeframe) if we settle on a single way to run more complicated sites with frontends and backends (one PaaS to rule them all, if you will) then maybe a library would run an instance and host sites that way. It's within their mission and I'd welcome the use of my taxdollars to do that - in a sense they do, as they put many of their collections online.
Where in the namespace is a more interesting question. I host my own domain because I want to trust as few things as possible (I give up and trust the root servers) and I need to keep paying for it for the rest of my life (I'm OK with that - I've got about 50 years left, I can swing the $500+inflation.) If anyone ever got control of my domain, they'd be able to be me - by publishing as me, or accessing other services either through whatever OpenIDs I've got out there or just being able to use 'recover password' emails. That bothers me - there's nothing I can do but keep my domain active, but I want those parts of my domain to die with me. In that sense, it's not like real estate - no one thinks that the person who lives in my old house is me. If URLs had a temporal notation this would be less of an issue.
Remember the mantra - "Cool URLs don't change". If the root of the namespace relies on someone being alive and active, then URLs are going to change unless they're passed off to another institution designed to outlive its founders. This isn't impossible - there are plenty of endowments and family foundations, but I think it's asking too much for every person to start a foundation and endow it with enough money to last "forever" just to keep their webpages online.
Instead, we should solve this once. Here's one way, but there are certainly others: get a library to serve as a domain name registrar. I'd probably strike it as a deal that once you stop paying for it, it's done and it doesn't point to anything but their own servers. There'd be some window at the beginning where you can upload material, or you can add to a live "backup" while your domain is active, and once it ceases to be active, that's the site you're left with.
We'd make a policy change at the ICANN level to say this registrar can buy at a discounted wholesale for any domains it's holding in trust - basically, we just need to be sure that the registry is fairly compensated for the cost of answering queries at the root level to point back to the library, and for the initial bit flips to say "this domain is in trust" but that's not strictly necessary. We could fund it through taxes, or probably better a service fee associated with active registrations because this crosses too many governments.
(Another approach to solving a lot of this is with a content-addressable system and just scrap the central registry. It leaves out Aaron and everyone else who's already died, but if it was a better solution for everyone going forward, I don't think Aaron would mind)