No, they don’t.
Self-store containers are not prone to either condensation or rising damp — it’s a myth that they are! So provided you don’t take anything that’s already damp into your container, everything will stay dry.
Washing machines and dishwashers that haven’t been properly emptied can cause problems. If you are using your container for white goods, make sure you’ve checked everything before it goes in.
There’s more about condensation here…