We do not have rabbits here.
It's either a cat, a squirrel, a possum, or a bird.
Something is sampling the barely emerging cucuмbers.
How shall I keep away such varmints? In your opinion(s)?
Cats are the least likely.
Rabbits would be my guess as well- I'll take your word for it that you don't have any, although I find it surprising.
You could add deer, chipmunks, and raccoons to the list as well. Do you have geese or ducks around? Goats?
I'm not sure there is much to do except to fence them in or park behind a tree with a firearm. There are products out there designed to repel vermin but I don't believe they work all that well, such as foul-tasting chemical products.
If you are seriously desperate you could try putting nets around the plants but it's tedious and certainly not guaranteed. Besides, the plants need to grow.
I'll ask my cucurbit-savvy sister if she has any suggestions.