>What would the sane way to go about it?
Plan for at least a week, if not two ahead. This depends on the number of people in the household.
>Is it really possible that we can run out of things like milk and diapers nowadays?
1. Someone correct me if I am wrong but cargo transports haven't stopped (ships, planes, trucks, trains). These things will keep coming in and stocking up the necessary stores.
2. People will stock up at once and then the craze will subside.
3. Just-In-Time works great when things go well, not so great when things aren't going well. Countries have warehouses stocked with (non-perishable?) food in case things don't go well.
>This is the first time I'm seeing this happen in my life :(
When this whole thing is over, go and volunteer in some third-world backwater shithole of a country and help them out. For many parts of the world, this blip of a "food shortage crisis" is the standard fare, along with a general lack of, well, everything.
Also, a good portion of the western population does not know how to cook, or has a very limited range of cooking, baking and food preparation skills. Something to consider.