The REST API is one way WordPress, and other applications, communicate with the server. One example is the block editor screen, which relies on this to display, and save, your posts and pages.

The REST API request failed due to an error.
Error: cURL error 28: Operation timed out after 10001 milliseconds with 0 out of -1 bytes received (http_request_failed)

WordPress Site Health Failure

To find out what was causing this I disabled all my plugins and then re-enabled them one by one.

In my installation I found that the error came back when I enabled the Check Email plugin