I was watching a very recent TV show, the FBI agent told the police that there is no way for anybody to block a live stream of a pedophile torturing a kid nor to trace it. Even when it's being broadcasted from within the same country.
What am I missing here? I can think of lots of ways to block and the website and trace the owner of the domain. Was it just fiction?
"Can't be blocked" is impossible. Even if you don't know the location of the source (say, they're using a censorship-resistant system such as a Tor hidden service), you can still perform a denial-of-service attack, though doing so would have substantial collateral damage (a successful DoS on a Tor hidden service will likely render the entire Tor network unusable).
If you can find the source of the stream, there are many ways of blocking it, of varying degrees of legality, ranging from sending a takedown notice to the relevant ISP, to BGP hijacking.
"Can't be traced" is a different matter. Censorship-resistant networks such as Tor or Freenet are designed to be difficult to trace: finding a Tor hidden service might require the cooperation of eight different countries' police forces (six for the Tor nodes passing data along, plus the investigating country and the one the server is in). Freenet is even harder: unless you're watching as someone uploads a file to the network, it's impossible to tell the original source of that file.
In the scenario described in the question, the use of Tor is unlikely: Tor's latency is too high and the bandwidth too uncertain for streaming video. Additionally, there's no way to have "partial" knowledge of someone's location: it's either all or nothing. Freenet can't have been used, since it's essentially a distributed file store rather than a network of servers; streaming video over Freenet is impossible.
As far as I can tell, you’d need to define where it would be “Live Streaming” from. For example, if someone used darknet (Onion/TOR) to stream, it would be very difficult since there are a lot of randomized layers. No one would be able to block a live stream since it would come from various sources (nodes). There could be no mechanism to have ISPs block their clients from viewing it either, as they’d have to block TOR protocols, and it does nothing if someone uses even more proxies to bypass the blocks.
There would be a mechanism to determine who had watched it, based on checksums, length of the stream, and other tidbits investigators could piece together, however that too becomes a touchy subject since they’d literally have to have all ISPs do some form of flow (netflow/jflow) and calculations.
Short answer is, they would be hard pressed to stop it without breaking laws, and violating privacy while doing so.
For police, requests like this are made to the ISP (which would be easy to ascertain), or the streaming or hosting service (also easy).
The problem is the timeframe. It would take time to coordinate with the ISP or service. It is going to be 'impossible' to turn off a website within an hour, for instance.
