I installed WordPress 5.3.2 and using the TwentyTwenty theme for this non profit I joined. I was trying to make a new Page and save but it kept complaining the REST API call failed and the page would not save.
The REST API call gave the following unexpected result: (404) Not Found The requested URL /wp-json/wp/v2/types/post was not found on this server.
Looked at Tools->Heath Check and noticed that I missed some php lib. After installing the libs that did not fix the error either.
Fix:
You likely changed the Settings -> Permalink
I had changed it to just post. Once I changed it to custom
index.php/%postname%
the API error went away and verified that the heath check passed too.
Resource:
https://wordpress.org/support/topic/url-rewrites-not-working/
Mitchell Fang's Coding 
This error is because URL rewriting is not enabled or misconfigured in apache.
An a2enmod fixed the error for me
The Fix explained here helped me to resolve 404 issues on my website. All 404 errors wiped off as soon as I changed the permanent structure to Custom. Thanks a lot. It has been a life saver.
Thanks! This helped!