That said, if we have a very liberal definition of the word "website" to include any successor technologies where a device can be used to request a document, given an identifier, that looks recognizably like your webpage, this is doable. What you really need is an institution that you can trust to keep existing and to keep the necessary upkeep of your website as part of its mission.
The main institutions I can think of that have lasted for 500 years unbroken are churches and elite universities. If you were able to convince the Pope to decree that the church should keep hosting your webpage in perpetuity, that would likely work, but persuading him of that sounds very challenging. That said, universities are used to accepting gifts with sometimes eccentric strings attached. The gift will probably need to be large; but I imagine a $1B donation to Harvard under a condition that they continue to host and update the page as needed would likely work. Getting that sort of money is quite hard, but tbh probably easier than coming with a way of guaranteeing that your direct descendents keep the webpage up.
It might not even take that big of an Endowment to get the University to do something like that. Universities are pretty good at Endowment (Annuity) math (because they have to be), University web hosting is still relatively cheap (easy access to low cost labor from "passionate" students, a DNS TLD that mostly can't just raise prices for arbitrary profit reasons) and no signs that it wouldn't be so in perpetuity. (Just keeping mind the risks of data loss of cheap labor.)
A quick search didn't find me an Annuity calculator that can calculate past 100 years (and I don't have the Excel fu to do it by hand because I'm not an accountant), but just experimenting with some numbers: let's say $25/month covers expected hosting costs and a tiny bit of funds for other web needs (maybe a pizza allowance for students) to cover that $25/month for a full century at a somewhat low expected annual growth rate of 1% you only need to start with at least $19k endowment today to cover the annuity. You probably don't want to start that small for sociopolitical reasons (to give them more reasons to abide by the terms of the annuity for the full length of it), but on the flipside you probably don't need anything at all close to a $1B dollars to do such a thing either.
You could start a trust with ~1M in assets and if you avg 8% growth a year (taking into account management fees) you'd have 1B in 90 years just through compounding.
Doable for some, even easy for some, but for the vast majority of people.... very hard.
I'm assuming this plan kicks off at death, so your trust wouldn't need to spend much to maintain itself.
The guts of the box carry a multi array of small, cheap computes's and a very large array of SSDs. Encased in material to protect from damaging scenarios.
The instructions on the outside of the box are diagrams, and text written in all current languages on earth.
A copy of Wikipedia/whatever archive is included. As well as your personal autobiography.
It figures out the healthiest way to stay dormant when not in use. And the healthiest way to use a subset of the hardware in the box to ensure a 500+ lifespan.
Cover plate for interfaces to be removed/opened to use.
If you give a university 1B for a 500 year commitment, that means you're paying the university $5,000 a day to keep that service up.
Let's not restrict this to "successor technologies." If we look to prior art, the traditional solution has been to deposit a copy of a work in a library. In that light, OP sounds like they're looking for the concept of a "family chronicle."
These are going to work long after other elements of technology change. The newest tech are going to change more often than the foundation.
It is sponsored by the Church of Jesus Christ of Latter-day Saints, which is almost 200 years old as an organization, which plans to last for a very long time; financially conservative (strong savings and effectively no debt), tech savvy, have strong traditions of decisions by consensus (or unanimity of leaders), and owns among other things a climate-controlled vault inside a granite mountain that has contained large amounts of microfilmed genealogical records (now digitized -- yay!). As family-oriented (ie, for preserving multigenerational culture among other things) as they come. One goal is to learn Adam's/everyone's family tree, as far as possible, and keep it forever. There are of ~ 17 million members (I am one), and for many reasons it seems likely to stick around a long time (I gather that Tolstoy also thought so, when he visited, when it was much smaller :) . (They also own Brigham Young University aka BYU, and holds a twice-yearly conference whose contents are translated to like >90 languages, I think several dozen languages live during it, and heard or read by people in some 220 countries, if my rough memory serves.)
I guess the site could have some size limit on what can be added from one person's account, or attached to one particular ancestor, or something, but the web site with all features is free, and I don't know why that would ever change. I used to work there (among many) on some back-end stuff.
(Edits to the above for clarity, working there, and the Tolstoy & BYU mentions.)
Edit: I'm curious: what is the general nature of the content you would like to save for 500 years? Sounds intriguing. Would it be useful to others also? Another idea would be to put it in wikipedia and/or archive.org, if it really doesn't fit in familysearch.org .