It is possible to make a multiplayer game without using Unity’s Multiplayer Services… but having my own servers…? Using the Free version?

If your game is for PC, and you are about to publish it on Steam, you could use Steamworks.
I am using it for my first multiplayer game and it is working well. One of your players could be the host, and you just rely on Steam to pass the info.

I researched a bit as I am also thinking on publishing my game on other platforms, and I will probably build a different solution for Xbox and WIndows, using their capabilities.