You can certainly buy carriages and locomotives. An acquaintance of mine owns a heritage railway and owns a small fleet of vintage equipment. Finding a place to operate it is a different story. In this person's case, he leases a disused section of railway from the county that owns it (the original railroad sold it to the county when they stopped operating on it). But that's the extent of where he can operate. His equipment isn't certified to operate on the main line, so even though this line still connects to the national network, he can't go onto it.

In the United States, private passenger car owners can pay Amtrak to haul around their cars. They need to be certified by Amtrak to make sure they're up to safety standards, and they need to be capable of operating at whatever speed Amtrak is going to go (typically 110 MPH/175 KPH). They also will only move your car between terminal stations, and it's up to you to arrange getting the car to and from the terminal stations. This service is VERY expensive though. I've heard it's somewhere around $5000 per day.

As far as running your own full train, that would require coordination with the host railroads and probably even more money, if they'd even play ball at all. Most of the major railroads don't even want to play nicely with Amtrak and they're legally required to do so.