People will often say 'Take a seat', and it seems rude to refuse, but you don't want to go into long explanations. But still, you don't want to sit down. If they are just inviting you to make yourself comfortable while you are waiting for something, you can say 'Thanks', then take an interest in the view out of the window, or the pictures on the wall, or ask where the toilets are.
If you are expected to sit down to talk to someone, then it is more difficult. To avoid long explanations to strangers, you can just say 'I've got a back problem - it hurts to sit down'. Often they will then tell you about their own back pains, or the problems of some relative of theirs, and the difficult moment has passed. Sometimes they will ask more, or ask if you have tried some treatment or other. I make it clear that I don't want to go into it by saying something like 'I've tried most things, but it still hurts'. Personally I avoid going into explanations with strangers. I have tried it in the past, and found I just got puzzled looks.
Of course, things are different with relatives, friends and work colleagues. They are entitled to more of an explanation, and need to know that you are not being difficult without reason. It may be hard at first to explain things, but in my experience life is easier once you have done it, and you feel more relaxed.
Some sufferers have encountered hostility or refusal to believe in their problem, by someone who has never heard of it before. If it is someone whose opinion doesn't matter to you, just walk away. If it is someone who must be convinced - like your boss, for instance - then you need to provide them with evidence that can't be ignored. The best evidence for them is a note from an orthopedic surgeon about your condition. Failing that, print out relevant pages from this web site, such as the paper by Maigne. This at least shows that coccydynia is a serious condition suffered by hundreds of people.
If your doctor doesn't take your pain seriously, this is a real problem. Here is advice on how to deal with an unhelpful doctor.
Family and friends may or may not be supportive. Unfortunately some will take the line 'I can't see anything wrong with you - stop complaining'. Or even if they don't say it, it's clear that's what they think. This is very hard to cope with when you're struggling to carry on with your life despite the pain.