http://en.wikipedia.org/wiki/Binary-to-text_encoding
With very modest client software modification, you could MIME encode any arbitrary binary file then squirt out one line of base64 per "post". Or the venerable uuencoded format.
What would prevent abuse would be some manner of throttling in the protocol to stop both this technique of blockchain abuse or arbitrary file length abuse.
I'm not thinking of apps like distributing cracked bluerays which would be fairly ridiculous, but much smaller binary files like a GPG key or SSH or whatever. Yeah yeah cut and paste whatever, why not automate it if you can. Click this menu option to send an extremely small file as a binary rather than screwing around with cut and paste.