I like Hong Ooi's answer. All I want to add is that time series forecasting often works in practice because things tend not to change much in the short term. For example, I don't know whether it will rain today. But it's sunny now, and it is very likely to be sunny 15 minutes from now. Or if you want to forecast the number of tourists visiting the country for next month, it will probably be roughly the same as last August. The number of tourists is increasing over time, and a forecast might take that trend into account, but even if it doesn't, it will still be pretty accurate.
We have to make these kinds of predictions all the time in our everyday lives. The only data we have, by definition, is what happened in the past. The best (and only) forecasts we have are the ones which predict the future based on the past. Time series methods are just a formal way of doing this. And ultimately, even when you run the most sophisticated models, the answer to the question "what will happen tomorrow?" often seems to be "roughly the same as what happened yesterday."
Your question is: what things cannot be predicted from time series? Many people would say the stock market. It's true that I can't predict tomorrow's closing prices. But in the very short term, the stock market is predictable, as the trader Jesse Livermore discovered at the beginning of his career (described in the marvellous book Reminiscenses of a Stock Operator.) Essentially, he was allowed to bet on minute-to-minute stock prices displayed on a ticker tape, and found that he could predict them accurately enough to make a steady income. But when he moved to Wall Street, he found that he could not trade real stocks quickly enough to exploit his knowledge, and went bust.
However, some things, even in the short term, are completely random. For example, there is no way that I can use time series to predict what the result of a coin flip will be, even if I've already flipped the coin many times. If you think about it, this is precisely the "gambler's fallacy". We have a very strong feeling that we can predict the near future based on the past, and it fails us when we are confronted with things which don't depend on the past. But the reason why we have this feeling is precisely because, most of the time, it's correct.
What things in real life behave like this? Maybe earthquakes? Supposedly you can predict how frequently they will happen, but not exactly when.