There are many definitions, but you can basically "feel" product-market fit when the market pulls product out of the startup (as defined by Marc Andreessen in the only thing that matters). Customers are banging on your door to get the product and this is your first API performance/load test - usage is growing as fast as you can add more servers.
Your product does not have to be great. it has to barely work and solve the market problem. The market does not care about how good the team is either, as long as it can produce a satisfactory product.
There is only one way to strive for this holy grail that is called product-market fit. Doing customer development. I always imagine the market as one circle, and the product as another. Finding product-market fit is a process in which you either move the product to overlap the market or vice-versa. You may pivot (change product or market) until you have fit.
You can always feel when this isn't happening: usage isn't growing, page views are meh, sales cycle takes too long, deals never close and word of mouth isn't spreading. Basically customers, aren't quite getting value out of the product. Imagine the overlapping circles again: it's like your product circle is satisfying 1% of the wanted market circle and the other 99% are users in other markets that you're trying to get on board. They don't really see it because you're not solving their pain. The hard part is to finding more of the 1% types of customers. Again - customer development.
There's something more to say about this holy grail. I think that products that hit this landmark in their iteration also get an advantage from the market - if they satisfy 10 users so well, these users might begin to share their positive experiences with others. All of a sudden, customers become your sales people.
There is also a core metric to test if you're unsure about this achievement. Ask customers a single survey question: "How would you feel if you could no longer use this product?" If >40% respond with "very disappointed" - you need to start scaling. If the percentage is lower, continue doing customer development and iteration until you hit the 40% benchmark.
I think you can also measure the fit of the product when a large percentage of users get to the "aha moment" - it's like meeting an attractive person and realising you're attracted to her/him. There's no misunderstanding or uncertainty about it. Replace the person with a product and the "aha moment" is the moment a user realises the value of a product.