My Internet connection was unreliable (there were a lot of timeouts) due to a partial outage at my ISP. This caused Acrobat DC to hang immediately upon opening it (I suppose because it couldn't reach some server). Switching to a more reliable connection fixed the problem immediately.

Adobe, you should make Acrobat more resilient against unreliable Internet connections. Not being able to open PDF files due to a spotty connection is unacceptable.