These lines kept cropping up in my system journal when I tried to access the server using ssh:

dbus[4865]: [system] Failed to activate service 'org.freedesktop.login1': timed out
sshd[7391]: pam_systemd(sshd:session): Failed to create session: Activation of org.freedesktop.login1 timed out
sshd[7388]: pam_systemd(sshd:session): Failed to create session: Activation of org.freedesktop.login1 timed out

The process list on the server looked fine. I could see dbus-daemon and systemd-logind processes and they were in good states. However, it looked like dbus-daemon had restarted at some point and systemd-logind had not been restarted since then. I crossed my fingers and bounced systemd-logind:

systemctl restart systemd-logind

Success! Logins via ssh and escalations with sudo worked instantly.