You can clone with
HTTPS or Subversion.
GitHub is a little weird: they want you to copy & paste your log/patch/file attachment to http://gist.github.com/, hit the Create Public Button and link to it from your issue by copying & pasting its URL.
As an example, like this: http://gist.github.com/455716.
For posting videos or screenshots of things not working properly you may find http://skitch.com/ (Apple Mac OS X only) or http://www.bleepingcomputer.com/tutorials/tutorial135.html (Windows Vista or later only) useful.
There's also some base64 encoder/decoders online that may be useful to get around this.
Option 3 - upload the document to Google Docs, make it public then paste the link to the document in the issue that you are creating.
This is a defect of git Hub and makes complex project management a mess. Guys when are you getting support for attachments released. Any plans?
This has nothing to do with BEurtle, the project to which this issue is attached. Please use https://github.com/contact to tell Github about any defects in their system.
Shortly, the correct answer is: you cannot! All the other comments are workarounds :(