The most efficient and expressive way is the following:
Use enums as Exits and use an EnumSet to store them. EnumSet is an efficient Set implementation that uses a bit field to represent the enum constants.
Here is how you can do it:
public enum Exit { North, West, South, East, Up, Down; }
EnumSet<Exit> set = EnumSet.noneOf(Exit.class); // An empty set.
// Now you can simply add or remove exits, everything will be stored compactly
set.add(Exit.North); // Add exit
set.contains(Exit.West); // Test if an exit is present
set.remove(Exit.South); //Remove an exit
Enum set will store all exits in a single long internally, so your code is expressive, fast, and saves a lot of memory.