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 overwritten 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.
Managing stocks at the parent product level will share availabilities between variations (i.e. if one variation is booked, the other ones will be unavailable aswell for these dates).
On the other hand, managing stocks at variation level means each variation have its own availabilities. And if one variation is booked, the other ones will still be available for these dates.