How can I translate a string without using the strings.xml resource file?

What you are looking for is an on demand translation. There isn't any free solution afaik. The best way for you is probably doing the translation on your server side and retrieving the correct string to the device.

You can call on your Android application the active lenguage with

Locale.getDefault().getDisplayLanguage();

And you can pass this value as parameter to your server.

Here, with a switch, an if or any other method, you can return the translated value of the string.

Hope this helps

Edit

Let's imagine you are calling the string from your server with something similar to

http://myurl.com/getMyStrings

Now you can add an input parameter to your call like

http://myurl.com/getMyString?lng=myLenguage

the myLenguage should be the locale result.

Now, server side, you can add some code. for example:

switch(myLngParam){
  case "English":
       return "My english string";
  case "Italiano":
       return "La mia stringa in italiano";
  case "Español":
       return "something else";
  default:
       return "default";
}

This way, based on your locale, you can have a different string.

NOTE: Obiouvsly this is an example, I don't know how you are calling the server but in any case for this goal you should add a parameter with the locale result