The most robust answer (i.e. that captures the intent of what you're trying to do while causing the fewest bugs) would be:

Object.size = function(obj) {
  var size = 0,
    key;
  for (key in obj) {
    if (obj.hasOwnProperty(key)) size++;
  }
  return size;
};

// Get the size of an object
const myObj = {}
var size = Object.size(myObj);

There's a sort of convention in JavaScript that you don't add things to Object.prototype, because it can break enumerations in various libraries. Adding methods to Object is usually safe, though.

Here's an update as of 2016 and widespread deployment of ES5 and beyond. For IE9+ and all other modern ES5+ capable browsers, you can use Object.keys() so the above code just becomes:

var size = Object.keys(myObj).length;

This doesn't have to modify any existing prototype since Object.keys() is now built-in.

Edit: Objects can have symbolic properties that can not be returned via Object.key method. So the answer would be incomplete without mentioning them.

Symbol type was added to the language to create unique identifiers for object properties. The main benefit of the Symbol type is the prevention of overwrites.

Object.keys or Object.getOwnPropertyNames does not work for symbolic properties. To return them you need to use Object.getOwnPropertySymbols.

var person = {
  [Symbol('name')]: 'John Doe',
  [Symbol('age')]: 33,
  "occupation": "Programmer"
};

const propOwn = Object.getOwnPropertyNames(person);
console.log(propOwn.length); // 1

let propSymb = Object.getOwnPropertySymbols(person);
console.log(propSymb.length); // 2

Native deep cloning

It's called "structured cloning", works experimentally in Node 11 and later, and hopefully will land in browsers. See this answer for more details.

Fast cloning with data loss - JSON.parse/stringify

If you do not use Dates, functions, undefined, Infinity, RegExps, Maps, Sets, Blobs, FileLists, ImageDatas, sparse Arrays, Typed Arrays or other complex types within your object, a very simple one liner to deep clone an object is:

JSON.parse(JSON.stringify(object))

const a = {
  string: 'string',
  number: 123,
  bool: false,
  nul: null,
  date: new Date(),  // stringified
  undef: undefined,  // lost
  inf: Infinity,  // forced to 'null'
  re: /.*/,  // lost
}
console.log(a);
console.log(typeof a.date);  // Date object
const clone = JSON.parse(JSON.stringify(a));
console.log(clone);
console.log(typeof clone.date);  // result of .toISOString()

See Corban's answer for benchmarks.

Reliable cloning using a library

Since cloning objects is not trivial (complex types, circular references, function etc.), most major libraries provide function to clone objects. Don't reinvent the wheel - if you're already using a library, check if it has an object cloning function. For example,

• lodash - cloneDeep; can be imported separately via the lodash.clonedeep module and is probably your best choice if you're not already using a library that provides a deep cloning function
• AngularJS - angular.copy
• jQuery - jQuery.extend(true, { }, oldObject); .clone() only clones DOM elements
• just library - just-clone; Part of a library of zero-dependency npm modules that do just do one thing. Guilt-free utilities for every occasion.

ES6 (shallow copy)

For completeness, note that ES6 offers two shallow copy mechanisms: Object.assign() and the spread syntax. which copies values of all enumerable own properties from one object to another. For example:

var A1 = {a: "2"};
var A2 = Object.assign({}, A1);
var A3 = {...A1};  // Spread Syntax