Unique one off items being double sold

I've installed zen cart 1.3.8 and love it, however, I'm having a major issue with it.
I sell one off unique items.
My problem is two or even more people can have it in their cart at once, and they can both/all complete the checkout process with that same one off item creating a refund nightmare for me.

One person returned to a 4 day old cart with an item still in it and successfully checked out the order 3 days AFTER it had sold to someone else!

Basically I really need zen cart to place a out of stock status on an item the moment an item is placed in a cart. Is there any way this can be done? (even if it means a coding change)

I'm aware of abandoned cart issues, so my next question is, is it possible to set a time limit on a cart (say 24 hrs) and after that time it just expires putting the items in it back to stock?

Thank You for any help given.

What's your setting for admin>configuration>stock>Allow Checkout

Allow customer to checkout even if there is insufficient stock

 true
 false

It should be false if you're selling one-of-a-kind items.

Firstly, A HUGE Thank You to afo for helping me out on this one :clap:, even though it was a hand slap to the forehead moment for me when I saw the 'fix'

I had found 4 similar threads that did not provide the answer to the poor souls affected, so just for the record...

From Admin under Configuration-Stock set the following...
Check Stock Level = True
Subtract Stock = True
Allow Checkout = False

With these options set, two people could still add the same item to their carts, but only ONE of them is allowed to check the cart out with said unique item in it. The other slower customer will get the message in a easy to see red box "Products marked with *** are out of stock or there are not enough in stock to fill your order.
Please change the quantity of products marked with (***). Thank you"

Another thing I will mention is, if you have been using test accounts BEFORE you set the above parameters, log them off before you test again. For some reason my test accounts ignored the settings till I had logged them off and on again.

Again, a HUGE thanks to afo for taking the time to give me the answer! I sure had a sleepless night thinking I had to go looking for another shopping cart system to use and throw away the the month of work I'd already done!