Sure. What you want is a PBX. VoIP is just a way to carry voice traffic. On the IP side, the number of conversations is only limited by the size of your Internet connection.
If you don't already have a PBX, look at Asterisk, a virtual Linux based PBX. Its IP connection is an Ethernet card. For the individual phone users, if you want to use existing phone wiring, the company that gives away Asterisk also sells adapter cards for regular phone extension lines, or you can run your PBX over your office lan and use individual IP phones or IP terminal adapters at each desk.