It's not as nice as the ?. operator, but to achieve a similar result you could do:

user && user.address && user.address.postcode

Since null and undefined are both falsy values (see this reference), the property after the && operator is only accessed if the precedent it not null or undefined.

Alternatively, you could write a function like this:

function _try(func, fallbackValue) {
    try {
        var value = func();
        return (value === null || value === undefined) ? fallbackValue : value;
    } catch (e) {
        return fallbackValue;
    }
}

Usage:

_try(() => user.address.postcode) // return postcode or undefined 

Or, with a fallback value:

_try(() => user.address.postcode, "none") // return postcode or a custom string

2020 Update

JavaScript now has equivalents for both the Elvis Operator and the Safe Navigation Operator.

Safe Property Access

The optional chaining operator (?.) is currently a stage 4 ECMAScript proposal. You can use it today with Babel.

// `undefined` if either `a` or `b` are `null`/`undefined`. `a.b.c` otherwise.
const myVariable = a?.b?.c;

The logical AND operator (&&) is the "old", more-verbose way to handle this scenario.

const myVariable = a && a.b && a.b.c;

Providing a Default

The nullish coalescing operator (??) is currently a stage 4 ECMAScript proposal. You can use it today with Babel. It allows you to set a default value if the left-hand side of the operator is a nullary value (null/undefined).

const myVariable = a?.b?.c ?? 'Some other value';

// Evaluates to 'Some other value'
const myVariable2 = null ?? 'Some other value';

// Evaluates to ''
const myVariable3 = '' ?? 'Some other value';

The logical OR operator (||) is an alternative solution with slightly different behavior. It allows you to set a default value if the left-hand side of the operator is falsy. Note that the result of myVariable3 below differs from myVariable3 above.

const myVariable = a?.b?.c || 'Some other value';

// Evaluates to 'Some other value'
const myVariable2 = null || 'Some other value';

// Evaluates to 'Some other value'
const myVariable3 = '' || 'Some other value';

Js logical operators return not true or false, but truly or falsy value itself. For example in expression x && y, if x is falsy, then it will be returned, otherwise y will be returned. So the truth table for operator is correct.

In your case you could use expression customers && customers.orders && customers.orders.Length to get length value or the first falsy one.

Also you can do some magic like ((customers || {}).orders || {}).length (Personally, I don't like this syntax and possible garbage collection pressure as well)

Or even use maybe monad.

function Option(value) {
    this.value = value;
    this.hasValue = !!value;
}

Option.prototype.map = function(s) {
    return this.hasValue
        ? new Option(this.value[s])
        : this;
}

Option.prototype.valueOrNull = function() {
    return this.hasValue ? this.value : null;
}

var length = 
    new Option(customers)
        .map("orders")
        .map("length")
        .valueOrNull();

It's longer than all the previous approaches, but clearly shows your intentions without any magic behind.