Is a woman who lies to escape those who meddle in her private affairs and embarrass her with questions sinful? And what should she do with them in the future?

The basic principle is that spouses have an independent life in which no one interferes except with advice or consultation. One should not inquire about others' private matters without a legitimate interest. The questioner erred by lying, as lying is forbidden except out of necessity or for a benefit that cannot be achieved otherwise, provided it does not harm others. There is no obligation to inform those who ask about private matters. Instead, one can use tawriyah (equivocation) and ta'ridh (allusion), by saying words that carry more than one meaning, intending a factual meaning while the questioner understands another meaning. If the questioning persists and one cannot avoid answering, then tawriyah and ta'ridh are permissible without resorting to lying.