Out of stock items still viewable
Out of stock, Best Match and Sales History
MarketPlace Connect needs a stock level to list on eBay. If stock is 0 the item will be not be listed. The only exception is if the stock was previously >0 and the item was listed and later the stock level reduces to 0. In that case, the listing remains on eBay but items cannot be purchased. That is done intentionally because if later stock is received for that item, it will automatically get relisted (if set to enabled) so that the listing retains the sales history/best match ranking.
The alternative would be to end all listings that reduce to 0 stock and relist when stock increases but this doesn't build up any sales history or best match ranking.
If you want to end listings that have zero stock, filter the eBay quantity column in the grid to 0 and bulk edit the eBay status column to disabled. Then "save and send to eBay" - (white button at top).