Some great answers here, let me add my views here. My experience on this topic relates to seeing a mobile app grow into millions of users, a content site growing to millions of uniques and ecommerce site growing over 500k in a few months.
Before getting into actions. Let's consider what scaling means first. Scaling is the simple of action of taking something you're doing well (or that fits well to a set of customers), and multiplying it in size considerably.
That means that if you don't do something well to start with, and you start to multiply it, the result is a grand mess and a shit ton of lost money.
Having established that we need something that works well for a set of customers, how do we know when that's true?
Well there's several ways of going about this:
1. Ask them and let them tell you. Probably not a good idea. Nobody likes to be mean.
2. Check their behaviour and see if it matches their intent. Best way to go.
Depending on your product you can choose different metrics to gauge if you have fit or not. It depends on the type of business you want to scale.
Ideally, as scaling means spending a bunch of money into acquiring new users, what you want is an approximate understanding of the return you will get for your investment.
For this reason, revenue is always a great metric to watch. Even better, lifetime value.
In terms of behavior though, it's really all about retention here. If your users come back fanatically and can't seem not to live without your product on a regular basis, then you have high chance of fit. Look at your cohorts and how they behave. Look at time spent on product.
You can also survey them to support this data.
Once you get numbers that stick. There's your green light to scale.
Good luck with your project. If you'd ever need more information on how to execute all of the above, then simply give me a call and I'll help like with many other companies.