Memex is a WebExtension, so it works on all browsers that support that API, which is most of them.

But there are many ways you can do that architecturally.

Here's one off the top of my head that might work:

1. Set up a DNS server on a home server.

2. Run a cron job to automatically wget every URL checked by the DNS server.

3. Set all of your devices to your DNS server.

4. Set up a local installation of one of the thousands of open source search engines and set it to go through all of your pages once an hour.

5. Enjoy.