Managing out of stock items

Some products are available to order even if we are out of stock, others, when they are gone are gone. We have some seasonal items that are sold out in a particular size and no further stock will be available yet customers appear to add these to their cart. What setting have I missed to prevent this happening in such products?

On a product, in the Inventory tab, there is a field "Low stock activity", with options to 'Disable buy button', or 'Unpublish'.

We need it to automatically prevent buy when stock in a particular size reached 0. I think what you suggest is a manual process.

No it is automatic
If you select 'Disable buy button' then when the stock goes to Zero then the Buy Button is no longer Visible
If you select 'Unpublish' then when the stock goes to Zero then the Product will no longer be Visible