The way you store Cucamelon plants is to bring the Cucamelon plant into a cool unheated space.
Cucamelons also do well growing in containers and can easily be moved indoors when it gets too cold outside.
After the first few frosts you can bring the containers indoors and keep them indoors until next spring.
You could store the Cucamelon plants in the garage or unheated shed just to keep them out of the direct cold but being in an unheated space is okay for them and they will produce more Cucamelons next spring.
If you want to store some Cucamelons after you pick them then you can leave them out on the counter or place them in a sealed container in the refrigerator to make them last longer.