I personally use SSL on my personal site just because I like the idea that readers can be sure the content is what I've sent, rather than because the information is private / sensitive. So I'm not personally concerned because this doesn't allow anyone to MITM the connection, just read it.
Adding data of random length will make the attack more difficult, but won't defeat it entirely. This is called length hiding.
See http://breachattack.com/ for a more thorough explanation by smarter people.
A counter-measure then would be to scramble the user input in the page with a random key, and include this key into the page for de-scrambling using JavaScript. It will still efficiently compress the parts of the page which are not user-controlled.
breachattack.com suggests masking the secret with a per-request random key, but masking the user input would work too.
No thank you.
Is my server vulnerable to this?