Sometimes people want to convince you to do things that you don’t want to do, and which aren’t any of their business.
Sometimes people want to argue with you about politics, and aren’t willing to have the conversation end unless you convince them or they convince you.
It’s ok to decide you don’t want to have those arguments. It’s ok to unilaterally end that kind of conversation.
You don’t have to convince them you’re right. You don’t have to convince them that you’re right about the issue in question, and you don’t have to convince them that you’re right about not wanting to discuss it.
It’s ok to say no to conversations you don’t want to have about things that are entirely your business.
The tricky part is backing out of the conversation gracefully.
You can say “I don’t think this is the right time to talk about that,” which works well if you’re in a very social situation with lots of people, or at work, in the middle of class, or another situation where an argument or a personal conversation isn’t appropriate. You can say “I don’t talk about that/politics/religion at work.” Or there’s simply “I’ll tell you later,” and then never do. Only a really rude person would press someone to follow up conversation they tried to get out of.
Sometimes you can head off a personal question by answering very vaugly. If someone asks why you don’t go home for the holidays, you can say, “My family and I are estranged.” You don’t have to give details. If they press, the phrase “It’s complicated” should shut most people up. But no polite person would press a stranger or co-worker to divulge family details. If they are being rude, say, “That’s a personal subject” or “I don’t want to talk about it right now.”
It’s a bit about your attitude, I think. If you’re nervous or giggly, they may think they can pry more info out of you. Don’t smile, try to immitate a grownup person you admire for their authoritative voice, and look the nosy person in the eye, or at least the forehead. Then, continue the conversation. Ask the nosy person a question (not a nosy one), or talk about something else.
Sometimes this works; sometimes it doesn’t. It doesn’t work on people who are willing to manipulate the rules of politeness to keep you in the conversation. People who are willing to do that can pretty much always arrange things such that there’s no way to leave the conversation without appearing rude. This is very common as a high-pressure sales tactic, but it can come up other places too.
When someone does that, you don’t owe it to them to keep following those rules of politeness.
(Also, people who can’t make or fake eye contact still have the right to decide not to have a particular conversation).