Is it possible to embed animated GIFs in PDFs?

It's not really possible. You could, but if you're going to it would be useless without appropriate plugins. You'd be better using some other form. PDF's are used to have a consolidated output to printers and the screen, so animations won't work without other resources, and then it's not really a PDF.

You can use Tikz/pgfplots for creating animations in beamer. http://www.texample.net/tikz/examples/tag/animations/

I do it for beamer presentations,

provide tmp-0.png through tmp-34.png

\usepackage{animate}

\begin{frame}{Torque Generating Mechanism}
  \animategraphics[loop,controls,width=\linewidth]{12}{output/tmp-}{0}{34}
\end{frame}

I haven't tested it but apparently you can add quicktime animations to a pdf (no idea why). So the solution would be to export the animated gif to quicktime and add it to the pdf.

Here the solution that apparently works:

1. Open the GIF in Quicktime and save as MOV (Apparently it works with other formats too, you'll have to try it out).
2. Insert the MOV into the PDF (with Adobe InDesign (make sure to set Object> Interactive> film options > Embed in PDF) - It should work with Adobe Acrobat Pro DC too: see link
3. Save the PDF.

See this link (German)