Any words meaning a situation when you can't joke?

Some adjectives that come to mind are serious, grave, solemn.

"solemnity" can be useful as a noun, for example:

• There was no room for jokes due to the solemnity of the occasion.

The term is Serious.

In common usage if the current situation will not permit humor one might be told, "This is serious." or "This is a serious situation." Once stated the joker knows where he stands and what a reception he will get with his five minutes of Vegas. This will cover both the moment asked about and the subject. One who cannot understand a joke is called Clueless. This is a different idea since it is not that they are too serious to understand, they just don't. One may be too serious to enjoy or permit a joke but they get it even if they would rather not.