All perfect praise be to Allaah, The Lord of the Worlds. I testify that there is none worthy of worship except Allaah, and that Muhammad is His slave and Messenger. We ask Allaah to exalt his mention as well as that of his family and all his companions.
It is not permissible for a woman to travel except with her husband or Mahram (i.e. unmarriageable relative) as illustrated by the Sunnah of the Prophet . For more benefit in this regard, please refer to Fatwa 90355.
If a woman has no husband and no Mahram and she is obliged to travel for a necessary matter, like being exposed to death otherwise, or due to a difficulty which she cannot bear if she does not travel, then she may travel with a safe company.
As regards going out when not travelling, then it is not an obligation for her to have a Mahram or husband when going out. It is permissible for a woman to go out without a husband or Mahram.
In principle a woman is required to stay in her own home and not go out unless she has to. Besides, when she goes out, she should observe all religious requirements, like wearing Hijab and the like.
For more benefit, please refer to Fatwa 84377.
Allaah Knows best.