I commented this, for my situation:
I have this same issue. I’m on an Ubuntu-server, but I’ve simply
pointed my host-file to point to the IP. I must admit I’m not 100%
sure how the restful API works, if that has any effect or not. I’m on
WP version 4.9.8, – and have the problem when I activate the
Gutenberg-plugin.
It seems that my .htaccess-file wasn’t writable (I saw that it wasn’t in the bottom of the ‘Permalinks’-page, where it said that it was a bit sad). When I added the .htaccess-file manually and uploaded it, with below-written content, then it started worked.
<IfModule mod_rewrite.c>
RewriteEngine On
RewriteBase /
RewriteRule ^index\.php$ - [L]
RewriteCond %{REQUEST_FILENAME} !-f
RewriteCond %{REQUEST_FILENAME} !-d
RewriteRule . /index.php [L]
</IfModule>