The data was about 55TB of compressed HTML last I looked, so that's about 70 r5a.24xlarge instances, each going for $5.424/hour, so about $350/hour or $250K/month. That's not cheap, and definitely not something you'd put on your personal credit card, but it's well within the range of a seed-funded startup. Sizes may vary a bit depending upon the exact index format, but that should be a rough ballpark. With batch jobs being so cheap, you could experiment a bit with your own finances and then seek funding once you can demonstrate a few queries where your results are better than Google. If you actually have a credible threat to Google, you'll have investors breathing down your neck, because it's a $130B market.

API access to either the unranked or ranked index in memory wouldn't do anything useful, BTW. To have a viable startup you need something a lot better than Google, which means that you need algorithms that do something fundamentally different from Google, which means you need to be able to touch memory yourself and not go through an API for every document you might need to examine. Remember, search touches (nearly) every indexed document on every query - if you throw in 200ms request latency for 4B documents your request will take roughly 25 years to complete.

Knowledge Graph is already public - it was an open dataset before it was bought by Google, and a snapshot of its state at the point Google closed it to further additions is still hosted by Google:

https://developers.google.com/freebase/

(It's only 22G gzipped, too - you can download that onto a personal laptop.)