It is possible. I've been running my own for a year, but it takes quite a hefty server at least with Synapse, and it's not cheaper compared to the quite nice 5 euros a month pricing here.

If you want to give it a try, this is what I used: https://github.com/spantaleev/matrix-docker-ansible-deploy

It always has been, since the beginning. This is purely convenient hosting for those not inclined to self-host.