The subthread Re^2: [OT] Source code repositories prompted me to look at Gitlab's website. They do prominently advertise their Enterprise Edition in the center, with a "pricing" button, but surrounded by a download button ("Download and install the open source GitLab CE in 2 minutes", it even looks like you can install from source) and a sign up button for "Free hosting for private repos" (the sign up page says "In the future we might charge for projects over 5GB", but from your OP it seems unlikely that you'll hit that limit).

Their using proprietary software for hosting doesn't change the license you use for the code you host there, so, while you're looking for a more appealing hosting facility (perhaps one of the monks hosts a gitprep(*) instance and lets you in), at least as a preliminary solution they don't seem downright appalling :-)

(*) Update 2016-05-09: Link rot, now https://github.com/yuki-kimoto/gitprep / http://gitprep.yukikimoto.com/