The Places API is in "Legacy" status as of March 1. From the developer docs (only present in English version):

This product or feature is in Legacy status and cannot be enabled for new usage.

The Places API has also been replaced, by the Places API (new). If you still want to use the old Places API, or if you are still having the issue even on the new Places API, you can manually activate the old one by going to this link. You might also need to edit the API Key API Restrictions and add the Places API to it. This will fix your issue.