Possible to access MVC ViewBag object from Javascript file?

Not in a JavaScript file, no.
Your JavaScript file could contains a class and you could instantiate a new instance of that class in the View, then you can pass ViewBag values in the class constructor.

Or if it's not a class, your only other alternative, is to use data attributes in your HTML elements, assign them to properties in your View and retrieve them in the JS file.

Assuming you had this input:

<input type="text" id="myInput" data-myValue="@ViewBag.MyValue" />

Then in your JS file you could get it by using:

var myVal = $("#myInput").data("myValue");

in Html:

<input type="hidden" id="customInput" data-value = "@ViewBag.CustomValue" />

in Script:

var customVal = $("#customInput").data("value");

In order to do this your JavaScript file would need to be pre-processed on the server side. Essentially, it would have to become an ASP.NET View of some kind, and script tags which reference the file would essentially be referencing a controller action which responds with that view.

That sounds like a can of worms you don't want to open.

Since JavaScript is client-side, why not just set the value to some client-side element and have the JavaScript interact with that. It's perhaps an additional step of indirection, but it sounds like much less of a headache than creating a JavaScript view.

Something like this:

<script type="text/javascript">
    var someValue = @ViewBag.someValue
</script>

Then the external JavaScript file can reference the someValue JavaScript variable within the scope of that document.

Or even:

<input type="hidden" id="someValue" value="@ViewBag.someValue" />

Then you can access that hidden input.

Unless you come up with some really slick way to actually make your JavaScript file usable as a view. It's certainly doable, and I can't readily think of any problems you'd have (other than really ugly view code since the view engine will get very confused as to what's JavaScript and what's Razor... so expect a ton of <text> markup), so if you find a slick way to do it that would be pretty cool, albeit perhaps unintuitive to someone who needs to support the code later.

I don't believe there's currently any way to do this. The Razor engine does not parse Javascript files, only Razor views. However, you can accomplish what you want by setting the variables inside your Razor view:

<script>
  var someStringValue = '@(ViewBag.someStringValue)';
  var someNumericValue = @(ViewBag.someNumericValue);
</script>
<!-- "someStringValue" and "someNumericValue" will be available in script -->
<script src="js/myscript.js"></script>

As Joe points out in the comments, the string value above will break if there's a single quote in it. If you want to make this completely iron-clad, you'll have to replace all single quotes with escaped single quotes. The problem there is that all of the sudden slashes become an issue. For example, if your string is "foo \' bar", and you replace the single quote, what will come out is "foo \\' bar", and you're right back to the same problem. (This is the age old difficulty of chained encoding.) The best way to handle this is to treat backslashes and quotes as special and make sure they're all escaped:

  @{
      var safeStringValue = ViewBag.someStringValue
          .Replace("\\", "\\\\")
          .Replace("'", "\\'");
  }
  var someStringValue = '@(safeStringValue)';