Following Encyclopaedia Britannica (and Wikipedia), space is defined as "boundless three-dimensional extent in which objects and events have relative position and direction". In classical physics, space is one of the few fundamental quantities, meaning that it cannot be defined via other quantities because nothing more fundamental is known at the present.
Space is the framework in which objects are embedded and in which events take place. Thus, in order to describe anything in nature, we must begin by defining space: we start by setting up a set of coordinates which we will use in defining the location of each object (often this is so obvious that we don't even notice that we do that!), and then put the objects inside the newly defined space, ("consider a ball AT R=0..."), quantify the objects and the relations between them, etc. - before we can start thinking of the evolution of the system.
Interestingly, many of the laws of physics rely on the experimental fact that space has 3 dimensions. This means that we need 3 independent numbers to fully determine the location of each object. In special relativity, Einstein showed that space cannot really be separated from another fundamental quantity - time. Hence in relativity, the correct entity is "space-time", which has 4-dimensions (3 of space and 1 of time).
Even more interestingly, in general relativity, it is shown that space is not "fixed", but can be "curved"; it can "stretch" with time, in a very similar way to a rubber band. Consider, for example, a rubber band, with ticks every 1 cm. Now, if the rubber band is stretched, the distance between the ticks grows. However, you can only notice that from the outside; if you, and all your belongings are within the rubber band, everything will stretch as the rubber band stretches. Thus, there is no way for you to tell that the rubber band is actually stretching! For example, if you live in 1-d space that is stretching, and measure an object to be, say 1 meter before the stretching, since the object stretches with space, after the stretch you will still measure its size to be 1 meter!. The big bang theory shows that this is exactly what happens to the universe.