How can I download HTML5 videos?

Depends on the site. Firefox has context menus for saving html5 video, but they don't work if the site uses a customized player. For example, if you visit http://www.mozilla.com/en-US/firefox/video/, you will be able to right-click and select the Save video option.

On YouTube, that won't work. In this case, click the YouTube logo in the address bar, then click More information in the mini popup, then click the Media tab in the big popup dialog, then find the video and select it, then click Save As. Change the extension to webm before saving.

Alternatively, you can copy the address and load it on some download manager like DownThemAll or wget via terminal.

However, on YouTube is simpler to use Video Download Helper.

With JDownloader you can download almost every kind of content available on the web. Its up-to-date addon system allows to select available quality and filetype easily (video-audio/audio only).

Not the easiest way but it's one of the most effective.

EDIT: Just discovered youtube-dl command. Its GUI version is also available through a WebUpd8's PPA, but is not that powerful as its backend. As I said before, is one of the most powerful CLI tools to download videos from almost anywhere in the web.

It can be installed from WebUpd8's PPA:

sudo add-apt-repository ppa:nilarimogard/webupd8
sudo apt update
sudo apt install youtube-dl

Or if you don't want to add a PPA, install latest version:

sudo curl https://yt-dl.org/latest/youtube-dl -o /usr/local/bin/youtube-dl
sudo chmod a+rx /usr/local/bin/youtube-dl

Update package without apt:

youtube-dl -U

And if you want to keep its manual in a text file (recommended), type this:

man youtube-dl > "youtube-dl manpage.txt"
youtube-dl --help > "youtube-dl quick help.txt"

Now go to your home (~) directory and pick them up, so you can work with the reference in one hand and the terminal in the other.

A Simple Way:

Just right click the video frame that is playing the video, and select "Copy Video URL" (depending on the browser you use) and paste the link in the address bar, and the browser should ask you to save the file.

A Better Way:

In case you're fond of downloading the video you see, whether on YouTube (Mostly based on Flash) or any other site (in case of HTML5 videos), you have two options,

1. Using download manager and add video urls (HTML 5 Videos will give you direct URLs).
2. Using something like Greasemonkey script that will add a "Download" button underneath any video you see in YouTube (Preferred for Flash videos, as they'll not have direct links).

In case of download manager for Ubuntu, I'll recommend DownThemAll add-on for Firefox (default web browser for Ubuntu).

And for adding "Download" button in YouTube, first you'll need to install Greasemonkey in the Firefox and then, use this userscript to add that "awesome" button in YouTube. And good thing is that the this Greasemonkey script is available for other web browsers as well.

Update

The Greasemonkey script mentioned in above URL doesn't work, however, I'm currently using this firefox add-on which works just fine and adds a pretty download button below the video frame.

If you use Firefox, they have a downloader, and for some videos it might require the installation of a companion app, which is quite straightforward. For me it works with 99% of videos on pages I visit with the Fox, if there is a video on the page you click on the icon and it offers you several options including to convert from one format to another. Magic for me.