I've recently wanted to write a streaming download service like this recently. The great thing about your service is that although it's a proof of concept of a handful of neat tricks, the core functionality works perfectly and could be extremely useful. This makes it faster than ever to drop upload links in a chat room and effectively solves the problem in https://xkcd.com/949/.
Is the backend licensed as free software? It would be fantastic for admins to host their own version of this since it allows the concept to propagate without worry of shutting down.
Currently I haven't licensed the backend as free software; the primary reason for this is that I reuse quite a lot of components from a different project of mine (https://gemin.ee) and I haven't decided yet on whether I want to release those components or remove them as dependencies. Since I too am familiar with useful stuff suddenly disappearing I'll give you my word that I will release the source if I have to shut down. If I decide to share the source I'll post a link to it on the site itself, so you won't have to look for it ;).
On another note, I'd encourage you to write a streaming download service. Compared to file uploading - storing - downloading I found it fun to do but also quite non-trivial to accomplish. I am planning to have another go at it myself, but with a different language (Go, Pony or Elixir). When run alongside of each other with the same UI/frontend/api it would make a nice benchmarking mechanism :).
Is the backend licensed as free software? It would be fantastic for admins to host their own version of this since it allows the concept to propagate without worry of shutting down.