Does it require root access to the machine I have a user account on?
Yes, from the README:
> Bocker runs as root and among other things needs to make changes to your network interfaces, routing table, and firewall rules. I can make no guarantees that it won't trash your system.
Linux makes it quite hard to run "containers" as an unprivileged user. Not impossible! https://github.com/rootless-containers/rootlesskit is one approach and demonstrates much of the difficulty involved. Networking is perhaps the most problematic. Your choices are either setuid binaries (so basically less-root as opposed to root-less) or usermode networking. slirp4netns is the state of the art here as far as I know, but not without security and performance tradeoffs.