Is it possible to embed video into an email that will allow the viewer to watch the video within the email itself in a consistent and cross platform compatible manner?
Youtube videos can appear to be embedded when being accessed using GMail's web client, but no, it's email, there are effectively no content standards.
There is no method of embedding video in email that guarantees playback in all email clients.
The best current workaround is to use HTML5 video embedding, suggested in this blog post from Campaign Monitor. This allows you to specify a fallback image that prompts users of email clients without HTML5 video support to click through to a web page to see the video in their browser.
Here's the conclusion from their blog post:
Here's their sample of the code you'd need to include:
Sample taken from "HTML5 video in email: An update on client support for 2011" on the Campaign Monitor blog.