I'm curious about how many hands one needs before they start taking stats like bb/100 or ROI% into consideration as an indicator of success. For example, having a 100 bb/100 in say 150 hands probably just means you ran really well in a session or having a 100% ROI in 3 tournaments probably means the same thing. So at what point do you think these stats become relevant? 5000 hands for bb/100? 10000? What about tournaments and ROI? 30? 100?

I think you need 10,000 as a minimum for the stats to become relevant.

Unfortunately its not so simple, that after x amount of hands and tournaments you are going to know your exact winrate. As your sample gets larger, standard variation will get smaller, but its still going to be there. But there is a point, where you can say for almost sure, you are not a losing player. Like its more than 95% likely, you are winning, more than 99% likely and so on. These free tools can give you a good idea about poker variance:

Poker Variance Calculator | Primedope

https://www.primedope.com/tournament-variance-calculator/

For instance to take a simple example we can look at the variance in a classic 9 man SnG over a sample of 100. Lets say the buyin is 4,5$, the fee 0,5$, and we think, our ROI is 10%. This mean, that over the 100 sample we are expected to win 50$ on average, but the standard deviation is 77$, and our risk of losing is 28%. So clearly with this format and a 10% ROI a sample of 100 is not enough to overcome variance.

However if we play 1.000, now the expected profit is 500$, the standard deviation is only 240$, and our risk of losing is only 1%. So while we are still not always going to win exactly 500$, we are at least almost always going to win something, if we truly have that 10% ROI and play 1.000 9 man SnGs. But here comes the next problem. Because few of us have the patience to sit and grind 1.000 of exactly the same SnG event. We are going to mix it up and play a bit of this and a bit of that, and that will tend to blur the data even more.

Moreover games tend to develop over time, and so does your own playing style. So dont get to hung up on your exact winrate. If you have a solid strategy and play in soft games, you should be able to win. And therefore its more important to focus on game selection and for cash games table selection rather than "knowing I am a winner in this particular game". Sticking with the 9 man SnG example, even if you grind the same one all the time, your true ROI might be 20%, when the games are best and 0%, when they are worst.