The pedantic answer (though not necessarily the most useful one, depending on what you want to do with the result) is:
string.length() * 2
Java strings are physically stored in UTF-16BE encoding, which uses 2 bytes per code unit, and String.length() measures the length in UTF-16 code units, so this is equivalent to:
final byte utf16Bytes= string.getBytes("UTF-16BE");
And this will tell you the size of the internal char array, in bytes.
Note: "UTF-16" will give a different result from "UTF-16BE" as the former encoding will insert a BOM, adding 2 bytes to the length of the array.