They use Rust in the datacenter. The layer that actually writes bits to disk is written in Rust. I don't know why they haven't yet deployed a native desktop client that's a Rust core with a thin platform-native GUI wrapper.
Edit: at 30:05 in that video, he briefly talks about a forthcoming (as of 2017) "full redesign + rewrite of Dropbox's 'sync engine' in Rust." He claims it will use 10x less memory, and be 50x faster on key operations.