Duckduckgo seems like the obvious alternative search engine these days, but how much of their own indexing do they do? It seems from some casual googling (no, the irony of that is not lost on me) that they mostly rely on Bing or Yahoo as the backend.
Some thoughts:
Search engines should be a lot easier to build in 2017 than they were two decades ago when Google started out. We don't have to store the index in DRAM to avoid hard drive seek times like they did; we can use SSDs.
Having a good quality ranking metric could offset shortcomings in the size of the index. PageRank is okay, but we could do better by punishing sites that link to low-quality content.
Speaking of PageRank, the PageRank patent expires in a couple years, I believe. So, constructing ranking algorithms that build on PageRank will soon be an option.
It's hard to compete with Google and Bing in terms of hardware resources, but probably the minimum investment to build a useful search engine is probably at lot lower than what we might think.
A lot of the basic software infrastructure is probably already written. I don't know what the state-of-the-art in web crawlers is, but in 2017 I doubt that one would have to write one from scratch.
Maybe there's a way to build a shared/federated distributed index.
Running the project as a non-profit might be the best way to maintain trust of users. Not having any ads at all (like Wikipedia) might be a good way to one-up Google's previous policy of no ads/news on the front page.