How to SEO a Single-Page website

Single page websites are bad for SEO. Period.

Since search engines rank web pages and not websites you need to make sure each web page is tailored towards a specific topic so it optimized to rank well for that topic. Putting more then one topic on a page dilutes that page's value for each topic and thus hurts its chances of ranking for both topics. It gets worse as you add more topics to that page. It also isn't useful for users since they have to sift through content they're not interested in to find the content they are looking for.

Additonally, internal links are valuable tools for SEO. They're almost as good incoming links from external sites and their anchor text does have value. Having a single page website means you are missing out on them.

I was thinking of something like having google index the different sections as different pages. Is something like that possible?

No.