Assuming you are specifically talking about their apphosting services (aka not dedicated servers), then the answer is mostly no. I say mostly, because some services you can manage to install and configure without needing root access or anything else that's blocked on HBD appboxes.
As for your example of AudioBookShelf, I'm pretty sure it wouldn't be possible, although maybe you can download the source from github and mess around with it if you are so inclined. I haven't tried and I don't have experience with ABS either, but I assume it's not a possibility.