I tried to add my self-hosted email as an account in settings, "Online Accounts". This makes the account visible to Geary. But Geary times out waiting to connect. I have added this account several times, so I've triple-checked that the settings and password are correct. I am able to connect with the K-9 app using the same settings on the same device. So clearly the network isn't a problem.
I've seen this report that Geary is working after a recent update. My software seems to be up-to-date but it's still not working, even after removing and adding the account, and rebooting.
I was looking for any alternative and tried Evolution. It's not designed for mobile so it'll never be a good solution. But I notice it already knew about my account. And it has the same problem, timing out when it tries to connect to the server. This means the problem is in the system, not in Geary the app. That's good news, because Geary is practically unmaintained in its present form. It means it's some kind of system problem.