Why doesn't java.util.Set have get(int index)?

Because sets have no ordering. Some implementations do (particularly those implementing the java.util.SortedSet interface), but that is not a general property of sets.

If you're trying to use sets this way, you should consider using a list instead.

Actually this is a recurring question when writing JavaEE applications which use Object-Relational Mapping (for example with Hibernate); and from all the people who replied here, Andreas Petersson is the only one who understood the real issue and offered the correct answer to it: Java is missing a UniqueList! (or you can also call it OrderedSet, or IndexedSet).

Maxwing mentioned this use-case (in which you need ordered AND unique data) and he suggested the SortedSet, but this is not what Marty Pitt really needed.

This "IndexedSet" is NOT the same as a SortedSet - in a SortedSet the elements are sorted by using a Comparator (or using their "natural" ordering).

But instead it is closer to a LinkedHashSet (which others also suggested), or even more so to an (also inexistent) "ArrayListSet", because it guarantees that the elements are returned in the same order as they were inserted.

But the LinkedHashSet is an implementation, not an interface! What is needed is an IndexedSet (or ListSet, or OrderedSet, or UniqueList) interface! This will allow the programmer to specify that he needs a collection of elements that have a specific order and without duplicates, and then instantiate it with any implementation (for example an implementation provided by Hibernate).

Since JDK is open-source, maybe this interface will be finally included in Java 7...

Just adding one point that was not mentioned in mmyers' answer.

If I know I want the first item, I can use set.iterator().next(), but otherwise it seems I have to cast to an Array to retrieve an item at a specific index?

What are the appropriate ways of retrieving data from a set? (other than using an iterator)

You should also familiarise yourself with the SortedSet interface (whose most common implementation is TreeSet).

A SortedSet is a Set (i.e. elements are unique) that is kept ordered by the natural ordering of the elements or using some Comparator. You can easily access the first and last items using first() and last() methods. A SortedSet comes in handy every once in a while, when you need to keep your collection both duplicate-free and ordered in a certain way.

Edit: If you need a Set whose elements are kept in insertion-order (much like a List), take a look at LinkedHashSet.