A while back I looked into a similar setup and was frustrated with the
hacker-esque nature of these tiny machines. They typically don't come
with a case, sometimes not even with power, and getting a working Gentoo
setup was likely going to be an effort I didn't want to spend.
So I ended up buying an Intel NUC: basically a tiny main board with a
CPU in a small simple square case + ram (you pick) + a hard drive (SSD
or HDD, you pick). It has HDMI or VGA out, sound, a few USB ports and
on-board ethernet as well. Getting Gentoo up & running on it was
painless once I turned of UEFI (it makes my head hurt). Details on how
I did it all was here: http://danielquinn.org/blog/gentoo-on-the-intel-nuc/
It'll cost you more than a Pi or some of the others, but it's basically
a tiny, quiet, whole computer, so the hassle is probably greatly diminished.