I hadn't tried that when I saw your comment, but then I did and the problems persisted. First, I tried deactivating all plugins and installing Jetpack. Same error.
Then, with the plugins still deactivated, I changed the theme do Twenty Eleven and reuploaded the plugin (downloading it from the search function doesn't seem to be working). Same error.
It's strange, because the search function doesn't seem to have any problem (I tried installing three different plugins after and they all worked). Uploading also works fine. It's just Jetpack.
The only thing I didn't know of in the Troubleshooting page of Jetpack was /xmlrpc.php. I tried accessing that and I got exactly what I should: a perfect one-lined text saying "XML-RPC server accepts POST requests only."
See? I have tried pretty much everything. The server is slow, so it's not letting me reinstall WordPress from the Dashboard, and I'm afraid of losing content if I reupload my original WordPress folder (downloaded from the website) onto the previous ones.
Is there any other alternative? Is this an isolated problem?