Why does Chocolatey fail to install Elixir on Windows?
Technically it passes installing Elixir - note the lovely green ball in the image: 
The green ball is also a link. Follow this Gist link to see the install results. I also just ran it while writing this answer. It installs just fine on supported Windows versions.
Two things are at play here (well, really just one thing):
- Windows XP. Technically Chocolatey is not supported on Windows XP (Microsoft released it over 14 years ago and ended support for it April 2014). Perhaps we need to officially state that somewhere (I looked around and while we've stated it before, I don't technically see it in our documentation).
- Downloading from GitHub requires better ciphers. It's not Chocolatey failing, it's likely the ciphers built-in to Windows XP not allowing it to download Elixir. I see Elixir downloads from GitHub, so it's quite likely that's why it doesn't work for you. Choco does attempt to fallback to SSL v3 when TLS fails, however the fallback likely doesn't even work because GitHub may have disabled that due to POODLE (the article is from CloudFlare, but it is a great explanation). See Windows support for SSL/TLS and GitHub's article disabling Windows XP and recommending upgrade to newer versions of Windows.
Also, another thing worth mentioning - Chocolatey has more detailed logs. If you don’t want to go grab the log file, you can also use -dv in your switches and just copy the screen output. Should you want to use a GUI interface in Windows, we also have ChocolateyGUI.
I'd highly recommend you upgrade to a newer and more secure Windows OS. Everything at and beyond Windows 7/Windows 2008 is a much better OS than Windows XP/2003. You can't go wrong in Windows 7 or Windows 10. Windows 8, well… That’s another story for another day.
Should you decide not to upgrade for now, you can likely install Chrome/Firefox and download and manually install Elixir from GitHub.