How do I export an array with module.exports in node js?

You'd want to define a function which returns the randomized portion of the array:

module.exports = {
  getRArray: function() {
    var arrays = [];
    arrays[0] = 'array 0';
    arrays[1] = 'array 1';
    arrays[2] = 'array 2';
    arrays[3] = 'array 3';
    arrays[4] = 'array 4';
    return arrays[Math.floor(Math.random()*arrays.length)];
  }
};

Also you should embed the array into the function so it actually returns something.

module.exports needs to be an object.

Perhaps you're looking for something more like:

var arrays = [];
arrays[0] = 'array 0';
arrays[1] = 'array 1';
arrays[2] = 'array 2';
arrays[3] = 'array 3';
arrays[4] = 'array 4';
var r_array = arrays[Math.floor(Math.random()*arrays.length)].toString();

module.exports = r_array;

Please note that this code will only be run once, and that if you're hoping to get a different random value by executing the code multiple times, that you may want to set it up more like:

module.exports = function() {
  return arrays[Math.floor(Math.random()*arrays.length)];
}

so that the Math.random() operation happens over and over.

You have two files, one is array.js and the other app.js. Consider doing this in array.js file:

const Array= [
    'array 0',
    'array 1',
    'array 2',
    'array 3',
    'array 4']
   
exports.Array = Array 

Now in your app.js do the following:

const { Array } = require("./array")

const randomArr = Array[Math.floor(Math.random()*Array.length)]
console.log(randomArr)
 // I used randomArr  instead of r_array because of JS conventions.