24/7 is an abbreviation which stands for "24 hours a day, 7 days a week". It is an American way of saying full-time. In commerce and industry, it identifies a round-the-clock service, potentially including holidays and other days on which work is normally limited.
In BDSM the term is used to identify or describe a relationship that is full-time, with no 'breaks' or 'timeouts'. Thus there is no question of being "in scene" or "out of scene"; life is one continuous scene. It would normally imply a live-together arrangement, but this is not necessarily indicated. What is most important for this description to be accurate is that the parties involved do not make a distinction between being "in scene" and "out of scene" and they do not differentiate between time spent with each other and time spent away from each other, as far as their Dom/sub relationship is concerned.
For example, even though two people might not live together, the commitment between them still abided, including any rules and understandings they have agreed upon, or any contracts they have entered into.
There are no hard and fast rules: like any relationship, how it works is totally between the people involved and 24/7 means different things to different people. It rarely means chained-at-the-hip but will mean that they do not consider the D/s aspects of their relationship as something separate or apart from how they might otherwise relate. It is important that they are both/all happy doing whatever it is they do to whatever degree they do it.
Instead of 24/7, people might meet up only occasionally or might be living together full-time but only involved in occasional play and therefore would not consider their relationship to be 24/7 D/s. Of course, the translation is somewhat subject to the interpretation of the parties involved, but is generally accepted with the same respect as a vanilla marriage agreement.
24/7/365 is sometimes also seen and has the same meaning, the 365 implying every day of the year.
Among Goreans, 24/7 is called "lifestyle".