I am trying to get the remaining space of one's Google Contact.
This FAQ says every account has 20MB / 25000 contacts. But the People API does not have any REST resource to list one's storage.
How can I get the storage usage/space through People API?
This information is not available to People API nor the deprecated Contacts API.
In order to see how many contacts you can still add, you can check how many contacts you currently have via people.connections.list and deduct that from the contact limit (25,000).
If you think this functionality could be useful, I'd suggest you to file a feature request.