I dropped my phone one too many times. This time the screen gave in and shattered.1 Whether related or not, my power button soon failed after and my phone was unable to boot. It would get about a second into the boot process before restarting again, as if the power button was constantly being pressed. The behaviour was the same even in recovery mode indicating that it was likely a hardware rather than software problem.
A Google search revealed that failure of the power button is a common problem with the Nexus 5. I tried the easy solution of partially disassembling the phone to get at the button and clean it, its rubber pad, and the surrounding area with isopropyl alcohol. Unsurprisingly, it didn’t work.
In my search, I came across this awesome article by Gregor van Egdom. It has a great explanation and photos of how this particular button works and why it likely fails. His solution is to replace the button. So I went ahead and ordered a replacement from Ebay ($5 with shipping from the US (I could have gotten it cheaper but I wanted it now rather than later as my phone was unusable)).