A Pi is fine, but don't rely on the SD card for storage. Home Assistant is very write-heavy (since it logs state changes, etc), and will kill an SD card pretty quickly. You're better off using the SD card as a read-only boot device and then use an external drive for the actual system.
I connected a 1TB SATA SSD to one of my RPI machines with the Element 14 "PI DESKTOP - HAT AND ENCL". ($50) It's a nice RPI case with some extras.
See: https://www.youtube.com/watch?v=Gh7AFRIdGi8
Thanks, that's important information. What kind of Pi should I use? Got a Pi 1 laying around which could be nicely powered by my router, but I doubt that will be enough...
