How do I estimate the market size of a given market?

Good news! Startups really don’t need to do a ton of secondary market research because 99% of the time, you are inventing a new product category, so existing segmentation by analysts for established product categories probably won’t help you all that much.

That said, you still need to do a back of the envelope estimation on your Total Addressable Market (TAM) to make sure you aren’t looking at a lifetime cumulative TAM of, oh, $5000. There are ways to do this.

• Count human beings who may become end users. If you are working in the US, there are two awesome resources: The US Census, and the US Bureau of Labor Statistics. Both provide really good numbers that can help you come up with an order of magnitude estimation.
• Find out which analysts cover your industry, then Google their names and report names to find press releases referencing new reports that just came out. A lot of times key numbers will be cited in these press releases.
• Do a bit of web research around key words in the space you are interested in, and see if there are prominent articles citing other research sources that may yield numbers that re useful to you. For instance, you may want to figure out the behaviors of millennials on Instagram. A quick Google search with the keywords "percentage of millenials who post frequently on instagram" in January 2018 immediately yielded several hits with statistics on various behaviors. You can then look at whether the data sources are credible, and then take the numbers you can find and extrapolate and interpolate.
• If you are a student with library access, you may have access to a lot of excellent reports via your school’s subscription with top tier analyst firms such as Gartner, IDC, IBISWorld and the like. For example the MIT Library System provides access to electronic resources which include some level of subscription to many of these analysts.

Market sizing is more art than science and approximation is the only way forward - even analysts who do this professionally have to take educated guesses based on available statistics which may not exactly answer the questions you are trying to pose. To get started, just get going and collect a few numbers, and see if you can find a way to model the total addressable market by combining these numbers in creative ways.