I had thought that אפשר was closer to the social acceptability end of the spectrum than to the physical possibility end, so the English glosses provided here ('can' 'possible' etc) don't have their centre of gravity where I would have expected it (they seem nearer to questions of physical possibility than of social acceptability).
Could one say here something like it's 'not allowed | acceptable | permitted' ? Is אפשר somewhere between יכל and מותר ?
ll אי אפשר לעשות את זה means "It's impossible to do it" or "It's not possible...". מותר is allowed/permitted, and you can use מותר לעשות את זה to say that some action is allowed. The negation of מותר is אסור, which is used similarly.
If something is not socially acceptable, you could perhaps use נהוג or מקובל (negation with לא for both).