It's not preserving the original site, rather archiving snapshots of it, but my go-to place for saving my personal websites for posterity is the UK Web Archive[1] run by the British Library. Given that they have a legal responsibility to archive stuff in a retrievable format there's no worries about them running out of money and they'll have an incentive to keep things available to view after the tech evolves out of recognition.

[1] - https://www.bl.uk/collection-guides/uk-web-archive