On the admin product page, go to the “Inventory” tab, check the “Manage stock?” option and set the product stock quantity below.
If you only have one item to rent, and want the date to be disabled when booked once, set the stock quantity to 1.
The “Allow backorders?” option is useless here, as it is overriden by the plugin.
For variable products, you can either set these options on the parent product, or on each variation. Or make a mix of both possibilities.