In addition to YaCy and the varieties of Searx (both of which perform better for me than any of the commercial search engines), it's not even out of the question to do this yourself, if you're willing to start with the most recent Common Crawl dump and do some spidering in between releases. I don't recommend it, unless you want to learn for yourself why search engines often give such miserable results, but it's possible.
However, that's the issue, here. Can you self-host a search engine? Sure, if you want to maintain the storage to back it. That depends on how deep your pockets go...
I keep saying "no" to this sort of thing, for a variety of reasons.
I mean, I get it. The language-model people are exhausting, and their disinterest in copyright law is unpleasant. But asking an organization that doesn't care to add restrictions to a license that the companies don't read isn't going to solve the problem.