That's really where torrents shine. I don't think the mega extensions will allow you to resume a download after you've closed the browser.
Upon googling 'resume mega download' there are some tricks posted involving putting the computer to sleep then waking it up to continue the download. You could possibly try that method (use hibernate if you must have your computer 100% off).
- Use the latest official Mega extension.
- Start download.
- <Need to turn computer off>
- Without closing the browser put your computer into hibernate mode.
- <Some time later>
- Turn the computer back on.
- Don't mess with the browser.
- Hopefully the download resumes where it left off.
You should probably test to make sure this works for you before investing a lot of time on it.