Thanks eschwarz, it works a treat!
In the end I passed the path to the user database, and it all works now.
This is my file:
[Unit]
Description=Calibre Service
After=network.target
[Service]
ExecStart=/usr/bin/calibre-server \
--url-prefix /calibre \
--port=8080 \
--enable-auth \
--access-log=/any_path/.local/share/calibre.log \
--userdb=/home/me/.config/calibre/server-users.sqlite \
/Calibre
[Install]
WantedBy=default.target
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