How do I throw a 404 error from within a java servlet?

The Servlet API gives you a method to send a 404 or any other HTTP status code. It's the sendError method of HttpServletResponse:

public void doGet(HttpServletRequest request, HttpServletResponse response) {
    response.sendError(HttpServletResponse.SC_NOT_FOUND);
}

In your doGet or doPost method you have a parameter HttpServletResponse res

404 is a status code which can be set by:

res.setStatus(HttpServletResponse.SC_NOT_FOUND);