EDIT: FileReader should now be safe to use instead of involving Flash.

A cross-browser solution to read a local file and provide the base64 encoded content with Flash.
See the example to know how to use it.
The example provides a polyfill-like solution (based on this swf file and on the FileReader API when available).

You'll need to define in your JavaScript the below code:

var Flash = {
  getFileData: function(base64) {
    /* this function is mandatory to deal with the base64 for the Flash version (old browser) */
  },
  getButtonLabel: function() {
    return "Load your file";
    /* you can use this function if you want to customize the button label for the Flash version (old browser) */
  }
};

1. Do the changes you need in the mxml file (Flex);
2. Download the Flex SDK;
3. Unzip and open a console;
4. Run the following command:

$ ./bin/mxmlc FileToDataURI.mxml

You'll have some warning messages as below:

Chargement du fichier de configuration D:\flex_sdk_4.6\frameworks\flex-config.xml
RSL requis :
http://fpdownload.adobe.com/pub/swz/flex/4.6.0.23201/framework_4.6.0.23201.swz avec 1 basculement.
http://fpdownload.adobe.com/pub/swz/tlf/2.0.0.232/textLayout_2.0.0.232.swz avec 1 basculement.
http://fpdownload.adobe.com/pub/swz/flex/4.6.0.23201/spark_4.6.0.23201.swz avec 1 basculement.
http://fpdownload.adobe.com/pub/swz/flex/4.6.0.23201/sparkskins_4.6.0.23201.swz avec 1 basculement.
http://fpdownload.adobe.com/pub/swz/flex/4.6.0.23201/mx_4.6.0.23201.swz avec 1 basculement.

But it will work and you'll find the FileToDataURI.swf file in your working directory.

Thanks to Country/FileToDataURI for giving me the inspiration !

For the first run, it's going to be a bit long to load the swf file. Actually the browser will connect to Adobe to download the swz files above. It's doing it just once, then no more download.