What people actually do is write their long text, put it in an image, and tweet the image.

Here's a way to do both: image and multiple tweets http://talltweets.com/

I tried that. It doesn't work. The character countdown appears to be letting you put in a long URL, but it rejects the tweet upon posting. Specifically I tried putting the first section of "A Tale of Two Cities" into URL form :-P

Surely it has some hard cap, otherwise I can see this being abused (base64 dumps)

Not sure what the appeal of inline base64 would be, though, over linking to a hosted file. Twitter is about sharing text, and arbitrarily long URLs mean mom and pop can achieve longer tweets by just adding http:// and using dashes.

not really, considering it'll get truncated

Apparently they still count data: URLs towards the limit.