Can I construct a JavaScript object without using the new keyword?

You can use this pattern:

function SomeConstructor(){
   if (!(this instanceof SomeConstructor)){
        return new SomeConstructor();
   }
   //the constructor properties and methods here
}

after which you can do:

var myObj = SomeConstructor();

In addition to this (rather old) answer: you can use a module pattern to create an object:

function Person(name, age, male) {
  name = name || 'unknown';
  age = age || 0;
  function get() {
    return ['This person is called ', name,
            (!male ? ', her' : ', his'),' age is ',
            age].join('');
  }
  function setAge(nwage) {
     age = nwage;
  }
  return Object.freeze({get: get, setAge: setAge});
}
// usage
var jane =  Person('Jane', 23)
   ,charles = Person('Charles', 32, 1)
   ,mary = Person('Mary', 16);

console.log(jane.get()); //=> This person is called Jane, her age is 23
mary.setAge(17);
console.log(mary.get()); //=> This person is called Mary, her age is 17

Here's a jsFiddle for some Date functionallity I created using that pattern.

What's wrong with using the new keyword?

At any rate, it sounds like the best thing to do is read up on Javascript inheritance: http://javascript.crockford.com/inheritance.html