Normally the office machine running Calibre would be on an internal network connected to the internet through a router. The key to giving people outside the network access to the content server is to set up the appropriate port forwarding on the router. If you do not have the necessary privileges on the office network to do this it is unlikely that your IT people would let you do so. There are ways around it but they are not worth your job, assuming of course that you are not the boss.
If you can port forward, it should not be a problem. Change the port that the Content server runs on to a port about 1000, preferably one not assigned to an application. For examploe, have the content server running on, say, port 33333. Make sure you also set a username and password. Then on the router forward port 33333 to port 33333 on whatever machine is running calibre. Then any web browser can connect to your Calibre server by entering, say,
http://www.example.com:33333, or whatever the external ip address of your network is, also with :33333 appended. The machine must of course be on and have Calibre Server running.