The new REST API is turned on automatically as of version 4.7.0.

My site was hit, as far as hacks go this one wasn't too bad. They defaced the last post, the solution was to revert to an earlier revision and upgrade WP to version 4.7.2.

If we would have had auto updates enabled then this attack would have been prevented. So the takeway from this is make sure that auto updates are enabled.

Why did you disable auto-updates?

Because updates occasionally break my site

Only security updates (v0.0.X) install automatically. Version change updates (vX.Y.0) always require a site owner to explicitly install them.

If you are doing things in WordPress that break with a security patch, you need to re-examine what it is you're doing.