How can I make a really long string using IndexedDB without crashing the browser?

Storing a Blob will use a lot less space and resources as there is no longer a need for conversion to base64. You can even store “text/plain” objects as blobs:

var blob = new Blob(['blob object'], {type: 'text/plain'});
var store = db.transaction(['entries'], 'readwrite').objectStore('entries');

// Store the object  
var req = store.put(blob, 'blob');
req.onerror = function(e) {
    console.log(e);
};
req.onsuccess = function(event) {
    console.log('Successfully stored a blob as Blob.');
};

You can see more info here:
https://hacks.mozilla.org/2012/02/storing-images-and-files-in-indexeddb/

Chrome has supported this only since summer of 2014: http://updates.html5rocks.com/2014/07/Blob-support-for-IndexedDB-landed-on-Chrome-Dev so you cannot use this on older versions of Chrome.