There are two main ways:
After a funding round, the valuation of a company is whatever the post-money valuation of the company is, by definition. Here there is a market - it's the market of investors willing to invest money into the company, and the shares are worth whatever the investors were willing to pay for them.
Also, every 12 months startups are required to get a 409a valuation done [1]. Basically, they hire an appraiser (often an investment bank), and that appraiser comes up with a value for the company based on how they would value it if they were shopping it around for an acquisition. Usually this involves looking at comparable companies, previous funding rounds, financials, growth potential, etc. That's the value that stock options must be offered at.
It's usually not easy (or even possible) for employees to sell shares at that price, for a couple reasons. First, anyone buying shares in a private company needs to be an accredited investor, which means they need a million in net worth, $200K in income for the last two years with a reasonable expectation of making that again, or the ability to pass a financial knowledge test. Second, companies usually want to keep their cap tables small (for a variety of reasons: it makes corporate governance simple, it makes the company more attractive to new investors, it eliminates potential roadblocks to an acquisition, and they can't go over 400 shareholders without needing to report financials as if they were a public company anyway), and so the stock agreements of most companies have covenants preventing you from selling your shares except with company permission, which they will usually not grant. That makes it hard or impossible for you to sell 40% of your stake to cover the tax bill.
FWIW, if the company does have a liquid secondary market in its stock, you can do what's known as a "sell to cover" transaction, where you borrow the money needed to exercise the options, sell the shares immediately, and then pay back the money you borrowed to exercise the options, effectively just profiting the difference between strike price and market value. These are not uncommon; back when FANGs gave options rather than RSUs they were the most popular way to exercise stock options and sell them on the public market, because the transaction is simple, immediate, and has no risk to the employee.
[1] https://learn.angellist.com/articles/409a-valuation