The cloud givith and the cloud taketh

They don't literally mean no batteries. They just mean small batteries. The 50Wh battery in my (modern, efficient) laptop lasts about 18 hours for example.

You'd also have battery powered lighting.

The real challenge is heating and cooling. If you want to be able to keep your house a comfortable temperature, your food cool in the fridge, your food hot when you eat it... that's not easy to do with small batteries. But it can be done, e.g. with good insulation and by changing your habits a little (cook during the day, etc).

You can also, as it says in the article, use "non battery" storage. We already do that. For example lots of people keep hundreds of litres of hot water next to their house. That hot water can be used, for example, to keep warm overnight. You can also fill empty air space in your fridge with water - unlike air, which is instantly replaced with warm air every time you open the door, the cold water will stay in the fridge and help the fridge stay cold much much longer. Easily overnight.

Of course, you could also just use gas for all of that... but if one of your motivations is to avoid carbon emissions then that's off the table.