Praise be to Allah.
A woman may live alone subject to the condition that she is trustworthy and is not a woman of dubious character. With regard to a woman travelling without a mahram, it is clearly not allowed, as in the hadeeth narrated by al-Bukhaari (1729) and Muslim (2391) from Ibn ‘Abbaas (may Allaah be pleased with him) who said: The Prophet (peace and blessings of Allaah be upon him) said: “No woman should travel except with a mahram, and no man should enter upon her unless a mahram of hers is present.” A man said: “O Messenger of Allaah, I want to go out with such and such an army and my wife wants to go for Hajj.” He said: “Go out (for Hajj) with her.”
This is completely wise, because travel involves hardship and exhaustion, and because of her weakness a woman needs someone to help her and stay by her side. Things may happen to her that may cause her to lose control when her mahram is not with her. This is well known nowadays when there are so many accidents involving cars and other means of transportation.
Moreover, if a woman travels alone, that exposes her to temptation and men may make approaches, especially when there is a great deal of corruption. Someone may sit near her who does not fear Allaah, and he may tempt her to do something haraam.
If we assume that a woman is travelling alone in her car, she is exposed to other dangers, such as her car breaking down, or evil people ganging up on her, etc.
This makes its clear that Islam is the first of all systems to take care of women, protect their honour, respect them and regard them as precious pearls that must be protected from evil.
We submit to the command of Allaah and His Messenger (peace and blessings of Allaah be upon him), and we know that it contains complete wisdom and mercy, because Allaah only forbids to His slaves that which is harmful for them.
It is not correct to compare travel to a woman staying alone in a house in her own land, because there are more dangers in the place that a woman travels to. If a woman is in her own town, if anything happens to her or she needs someone to help her, she will find someone to help her. The fear of evil people attacking her is less when she is in her own town and her own house than when she is travelling.
And Allaah knows best.