• Arrays are indexed starting at 0. If your code actually did compile, when you ran it, you would get an out-of-bounds error.
• You don't need to call a String constructor (and you can't as per the error). When you assign x + " " + z, it creates a String for you because of your string concatenation. Because your array is non-typed, it would store non-string values if that's what you were actually passing in (you aren't), but if you were passing a non-string object to something expecting a String value, in most cases, it would implicitly call object.ToString() for you so you shouldn't have to worry about that anyways.

This will do what you want:

var  mapSizeX = 256;
var  mapSizeZ = 256;

function Awake () {
    mapArray = new Array(mapSizeX);
    for (var x = 0; x < mapSizeX; x++) {   
        mapArray[x] = new Array(mapSizeZ);
        for (var z = 0; z < mapSizeZ;) {
            mapArray[x][z] = (x+1) + " " + (++z);
        }
    }
    Debug.Log(mapArray[1][1]); //0,0 stores 1,1. 1,1 stores 2,2. etc.
}

But like Eric5h5 wisely points out, the c# multi-dimensional arrays are better anyways if they don't need to be resized.

Strings are structs and do not need to be initialized with new, and in fact cannot be. Unless the arrays need to be dynamically sized, however, it would be quite a lot faster to use this for actual 2D string arrays instead of using Arrays of Arrays.