I might (mis?)intepret "This is out of the question." as "You are not allowed to ask about this." Is that a possible meaning for a native speaker, or no?
For that you'd say (in an informal context) 'that's off limits', 'that's not on the agenda' or 'that's not on the table'. I don't know of a (short) way of saying it formally - you'd have to use some periphrasis like 'I'm not at liberty to talk about that' or 'that's not [up] for discussion'. Of course, a politician would say 'I'm glad you asked me that' and proceed to talk about one of the few things his party has done right.
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If it's not as strong as "not allowed to ask", does it imply that the person who said it doesn't want the person he's speaking to ask about something?
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I'd say "It's out of the question" is plenty strong in the sense that it completely rejects an idea or suggestion. There is not a sense that someone isn't allowed to ask the question or bring the topic up, but rather that there is simply no point in discussing the idea/suggestion further.
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