Shade is caused by some object blocking the sunlight. So, opposite of sunny in this case cannot be shady(?).
So, when there is no sun in the sky and therefore it is not sunny, then how should I describe the climate/area?
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The best adjective in your context would be "overcast." As defined by Wiki:
Overcast or overcast weather, as defined by the World Meteorological Organization, is the meteorological condition of clouds obscuring all of the sky.
An example of overcast weather:
"Cloudy" is fine, too. But if you want to convey that there is no sunlight at all, then this would not be the best option. You need to further describe it. Sky can be very cloudy, but this doesn't mean that there is no light striking the land because of the clouds, as clouds can very well be transparent.
You can, for example, see here:
You can read these definitions taken from ldoceonline:
Is there no sun in the sky because it is covered by clouds? Use "cloudy", "overcast", or "grey" (or "gray").
Or because it is night? If there are clouds, "cloudy" and "overcast" still apply. If there are no clouds (so you're trying to say "it would be sunny if it wasn't night"), use "clear" or "clear skies".
If it's raining (or hailing, snowing, etc.), then say that. It's implied that you can't see the sun when it's raining, unless you very specifically say so (a "sun shower").
The adjective "sunny" means "being with sunlight; having a lot of sunshine."
You may use "cloudy," "dark," "overcast," or "gloomy" as an antonym of the "sunny," depending on the context.
But I think the most appropriate antonym is "sunless."
The adjective "sunless" means "being without sunlight; having no sun."