The simple answer is no, you cannot make a typical computer into a PSTN-connected PBX without a new piece of hardware: an "FXO card". An analog modem cannot perform as a voice-service FXO as they are designed only to process data bits asynchronously (i.e., V.92 protocol) – that's why modems are much cheaper and more common than FXO cards. A LAN card's port doesn't accept analog signaling at all and cannot handle your telecom's line voltage either, so don't plug a phone line into it!
In my opinion, if you are dealing with fewer than four telephone lines coming into your premises, I would not build a PC-based PBX but rather buy a smaller embedded device to interface with it.
The Linksys SP3102, for example, will handle one phone line, and allow you to make or receive calls with it via a desktop SIP app.
Bought second-hand, it is cheaper than many quality FXO cards, which you would need to build an Asterisk or FreeSWITCH PBX yourself. I have never used a USB-FXO, but I doubt call quality or latency will compare favorably to a dedicated PCI card (remember those things?).
Also, the time to set up and debug Asterisk/FreeSWITCH to work reliably with any FXO card is substantial, whereas the Linksys box can do basic call processing by itself after configuring it via a web page. Later on, you can integrate it with a separate Asterisk server if you want more configurability or voicemail storage, etc.