calibre has no size limits, but e.g. Gmail has a 25 MB limit for attachments.
calibre can directly email to your Kindle (actually, it uses your email account as a relay, once you enter your username+password in the settings). But as AnemicOak said, there is no Chromebook version.
Although... you can use Crouton to install a Ubuntu chroot alongside ChomeOS, that will allow you to run native software rather than limiting yourself to cloud-based stuff. calibre would then work.
calibre-server must run on a supported computer, and will create a personal website served from that computer, which you can download books from via the browser.
Useful if you have another computer besides the Chromebook, and wanted to read books on your Chromebook (or download from the builtin browser on your Kindle)... but not really helpful in your case.
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In what way did Bookdrop not work out for you?
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Do note that you can always email things yourself, without involving Dropbox, although that may not be as convenient depending on your workflow.
See:
http://amazon.com/sendtokindle