The exchange is open from 9:30am to 4:00pm, but you and I are free to trade amongst ourselves at midnight on a Friday if we'd like

Aren't we also free to trade amongst ourselves during a market circuit breaker event?

Yes, but it's much, much slower to trade directly than through a market-maker in the exchange

how does one actually technically go about that? I have no plans to do it, just been curious about it over the past few months.

In my part of the world, tell your broker that you’ve done a deal with someone, tell that someone to tell their broker that they done a deal with you. The brokers should be able to report it to the exchange for publishing. Then they should see through the clearing and settlement. Of course your brokers might feel that your little deal is just not worth the hassle, and ask you to take your business elsewhere.

You ask your broker for the physical copy of the stock certificate. The buyer asks their bank for some physical cash. They hand you the cash, you hand them the stock certificates. Same as buying or selling anything else, basically.

Of course, for a transaction of a certain value, you wouldn't want to hold the cash or certificate, so instead you write out a contract and sign it at the moment of exchange. Which is the same as buying or selling something else (e.g. car, house) beyond a certain value.

> The actual prices of stocks are constantly changing 24/7/365

They are not. For any meaningful interpretation of "actual", the only price is the market price during trade hours. Speculative overnight agreements between private parties at agreed-upon values are contract agreements. The stock price doesn't change between market close and open.

What is the "market price"?

Is it the bid? Is it the offer? Is it the last trade? The VWAP?

The stock market only tells you where shares have traded and where the order book would trade them. There is no observable "actual price."

A useful way to think about the price is to imagine a theoretical fair value that is always changing, and a theoretical bid/offer that is always floating around the fair value.

This theoretical fair value is not posted anywhere. Some people give the name "price discovery" to the process of identifying where the theoretical fair value lies. With liquid stocks like AAPL on calm days, price discovery is a simple affair. With other assets, price discovery can be more opaque.

When the stock is trading steadily at high volume, then sure. The last trade is a great representation of the theoretical fair price. But in choppy trading, when there are dislocations between related assets and spreads are wide? The last trade is not representative of the whole picture.

Consider assets whose transactions must be reported to TRACE. If you are long, you have an incentive not to sell aggressively because a downtick will mark down the value of your position. So what's the fair price? Is it the last trade? Not necessarily, because that trade may not represent the current state of the market.

Also, there is generally some illiquid trading taking place pre-/post-market. And US futures trade overnight in other markets. And companies may be exposed to assets that trade outside of US trading hours (eg, refiners that have storage tanks full of crude, or companies that have currency exposure).

Perhaps I should've said "actual value" rather than "actual price", but my point otherwise still stands.