I’ve seen a lot of people do this when resurrecting old niche sites. The high-level approach usually involves grabbing all the snapshots from archive.org, stripping out their timestamped URLs, and consolidating everything into a local mirror. In practice, you want to:

1. Collect a list of archived URLs (via archive.org’s CDX endpoints). 2. Download each page and all related assets. 3. Rewrite all links that currently point to `web.archive.org` so they point to your domain or your local file paths.

The tricky part is the Wayback Machine’s directory structure—every file is wrapped in these time-stamped URLs. You’ll need to remove those prefixes, leaving just the original directory layout. There’s no perfect, purely automated solution, because sometimes assets are missing or broken. Be prepared for some manual cleanup.

Beyond that, the process is basically: gather everything, clean up links, restore the original hierarchy, and then host it on your server. Tools exist that partially automate this (for example, some people have written scripts to do the CDX fetching and rewriting), but if you’re comfortable with web scraping logic, you can handle it with a few careful passes. In the end, you’ll have a mostly faithful static snapshot of the old site running under your revived domain.