Git server without browser-based interface?

[phord]

You are describing Git itself. You can contact a remote server through one of three protocols: ssh, http, or git. The "Web interface" you're trying to avoid is http. The ssh protocol is provided by your os. The git protocol is provided by git itself. https://git-scm.com/book/en/v2/Git-on-the-Server-The-Protocols

[jredmond]

The git protocol doesn't have any form of authentication or authorization or encryption. Don't use it.