How to handle StackOverflowError in Java?

I'm not sure what you mean with "handle".

You can certainly catch that error:

public class Example {
    public static void endless() {
        endless();
    }

    public static void main(String args[]) {
        try {
            endless();
        } catch(StackOverflowError t) {
            // more general: catch(Error t)
            // anything: catch(Throwable t)
            System.out.println("Caught "+t);
            t.printStackTrace();
        }
        System.out.println("After the error...");
    }
}

but that is most likely a bad idea, unless you know exactly what you are doing.

You probably have some infinite recursion going on.

I.e. a method that calls itself over and over

public void sillyMethod()
{
    sillyMethod();
}

One to handle this is to fix your code so that the recursion terminates instead of continuing forever.

Take a look at Raymond Chen's post When debugging a stack overflow, you want to focus on the repeating recursive part. An extract:

If you go hunting through your defect tracking database trying to see whether this is a known issue or not, a search for the top functions on the stack is unlikely to find anything interesting. That's because stack overflows tend to happen at a random point in the recursion; each stack overflow looks superficially different from every other one even if they are the same stack overflow.

Suppose you're singing the song Frère Jacques, except that you sing each verse a few tones higher than the previous one. Eventually, you will reach the top of your singing range, and precisely where that happens depends on where your vocal limit lines up against the melody. In the melody, the first three notes are each a new "record high" (i.e., the notes are higher than any other note sung so far), and new record highs appear in the three notes of the third measure, and a final record high in the second note of the fifth measure.

If the melody represented a program's stack usage, a stack overflow could possibly occur at any of those five locations in the program's execution. In other words, the same underlying runaway recursion (musically represented by an ever-higher rendition of the melody) can manifest itself in five different ways. The "recursion" in this analogy was rather quick, just eight bars before the loop repeated. In real life, the loop can be quite long, leading to dozens of potential points where the stack overflow can manifest itself.

If you are faced with a stack overflow, then, you want to ignore the top of the stack, since that's just focusing on the specific note that exceeded your vocal range. You really want to find the entire melody, since that's what's common to all the stack overflows with the same root cause.