The oysters will be fine in the box they come in - it is insulated and recyclable! So once used break it down and put in your recycle bin.
Open the box up and sprinkle water on the hessian liner and leave a little open so air can come in and cool the oysters.
Store in a cool part of your house like the pantry, laundry or walk in wardrobe.
If you have a wine or beer fridge - they will last for days if kept at around 12 degrees. Regardless of where you put them, they will be fine for at least a week, if not 2 weeks, if kept cool around to 10 to 15 degrees - they just sit there waiting for the tide to come back in!
However also fine in your fridge, just won't last for weeks. Put them down near the veggie crisper where it's not as cold.
They are happy in the box they come in as well but like to be stored colder. Store in your fridge at 5 to 10 degrees. In your normal fridge put them down near the veggie crisper where it isn't as cold. Or put in your beer or wine fridge adjusting the temperature accordingly.
For all oysters...
Once shucked oysters need to be eaten them soon after shucking or refrigerated for consumption later. If shucking and refrigerating they should be stored in a container or covered with plastic wrap and will last 2 days maximum from shucking time.