The answer depends on the language you're using.
In C and C++, the keyword was NULL, and what NULL really was was 0. It was decided that "0x0000" was never going to be a valid pointer to an object, and so that is the value which gets assigned to indicate that it is not a valid pointer. However, it's completely arbitrary. If you attempted to access it like a pointer, it would behave exactly like a pointer to an object which no longer exists in memory, causing a invalid pointer exception to be thrown. The pointer itself occupies memory, but no more than an integer object would. Hence, if you have 1000 null pointers, it is the equivalent of 1000 integers. If some of those pointers point to valid objects, then the usage of memory would be the equivalent of 1000 integers plus the memory contained in those valid pointers. Remember that in C or C++, if a pointer no longer points to its object, that does not imply memory has been released, so you must explicitly delete that object using dealloc (C) or delete (C++).
Unlike in C and C++, in Java null is merely a keyword. Rather than managing null like a pointer to an object, it is managed internally and treated like a literal. This eliminated the need to tie in pointers as integer types and allows Java to abstract away pointers entirely. However even if Java hides it better, they are still pointers, meaning 1000 null pointers still consume the equivalent of 1000 integers. Obviously when they point to objects, much like C and C++, memory is consumed by those objects until no more pointers reference them, however unlike in C and C++, the garbage collector picks up on it on its next pass and frees up the memory, without requiring that you have to keep track of what objects are freed up and which objects are not, in most cases (unless you have reasons to weakly reference objects for example).