Bring extension cords inside before it freezes.
Yes, store them indoors if you can. Cold makes PVC insulation brittle, and the worst thing you can do is leave a cord coiled up outside in subzero temps, then yank it straight to untangle it next spring. That’s how you get hairline cracks that lead to shorts or shocks.
If you absolutely have to store them outside, keep them in a sealed bin or a shed that stays above 20°F. Coil them loosely—tight wraps stress the insulation when it’s cold. And make sure they’re dry before you put them away; water freezes and expands inside the jacket, splitting it from the inside out.
The real trick is to avoid sudden straightening. If a cord has been sitting in the cold, let it warm up in the garage or mudroom for an hour before you handle it. Cheap extension cords are not built for arctic conditions—treat them like the temperamental plastic tubes they are.
Future you will appreciate not having to replace a cord that looked fine but sparked the first time you used it.