Is it permissible for committed women to go on a recreational trip to an enclosed place without a man or a mahram, and what is meant by the hadith "He has permitted you to go out for your needs" and the hadith "It is not lawful for a woman to travel a distance of a day and a night except with a mahram", and is what is meant time or distance?

If the journey is within the country and is not considered travel, then there is no harm in it. However, if it is outside the country and is considered travel, then it is not permissible for a woman to travel without a mahram. As for the permission granted to the Mothers of the Believers to go out for their needs, what is meant by that is what does not lead to fitna (temptation/strife) and does not result in travel. Otherwise, a woman's travel has another ruling, which the Messenger of Allah, peace and blessings be upon him, clarified.