An example of using the Blob / createObjectURL web API: bundles tweets as a CSV and presents a download link;

tweets_blob.js

var csvDownload = function(){
  var t = $('p.TweetTextSize');

  var CSV = [];
  
  for (i = 0; i < t.length; i++) { 
     CSV.push('"' + t[i].innerText + '"');
  }
  
  CSV = CSV.join('\n');

  window.URL = window.webkitURL || window.URL;

  var contentType = 'text/csv';

  var csvFile = new Blob([CSV], {type: contentType});

  var a = document.createElement('a');
  a.download = 'my.csv';
  a.href = window.URL.createObjectURL(csvFile);
  a.textContent = 'Download CSV';
  a.style = "position:fixed; top:100px; left:100px; z-index:100000; background-color:black; color:white"
  a.dataset.downloadurl = [contentType, a.download, a.href].join(':');

  document.body.appendChild(a);
};

csvDownload();

Go to twitter.com, hit F12, and copy pasta this into the console, et voila, csv download link with the current tweet texts.