Step I
Open a tab and enter the following code int the nav bar :

about:cache

You should receive a web page informing you where exactly does the browser keep the cached data. It should be under the 'Disk cache device' title.

Step II
Access your target web page -for instance a youtube page.

Step III
Navigate to the cache directory. Sort the list view based on size or modified date; you should be able to see the file loading and getting bigger as data is streamed.

Step IV
Salvage that file to a safe location.

Step V - only if file gets deleted
This is the tricky one: sometimes FireFox deletes the cached file from the cache directory, despite the fact that you can replay it again. So it must be moved to some other place, of which i don't know. Any ideas?

However, you can use the following trick:

For Linux:
Open a console and navigate to the cache directory, something like:

cd ~/.mozilla/firerox/something/Cache

assuming your downloading cached file has the name of 'foo' (you will have probably something like 5AACF068d01 here ) run the command:

ln -t foo savemehere

This creates a hard link to that file, in the same directory. DON'T create a soft link as you will end up with a dead link when the file is deleted, so don't use the Gnome's file browser 'Make Link' for instance.

For Windows:
Same as above (using Ms DOS cd syntax and slashes, naturally).
In Windows the command for creating hard links is mklink /H. Yo can find a comprehensive article about hard links in Windows here

When the 'foo' file gets deleted you will be able to retrieve your 'savemehere' copy and do whatever you want with it.