I joined a company in California back in September 2007, I got a 2,000 options package. The stock price was 27 USD when I was hired. I left the company in July 2012. Options vested already but they were underwater back then. I think stock was around 17 USD. Today stock price is 30 USD. I went to my brokerage account and I don't see the stocks. Do I own them?
Stock options mean you have the opportunity to purchase the stock at a given price. They don't mean you actually own company stock, even after they vest, unless you exercise your options which involves paying the strike price specified by the option (though it's possible to do a cashless exercise with options that are above water). It's very unlikely that your company would allow you to exercise your options beyond 90 days after leaving. So unfortunately your options are probably expired and you don't own any shares of stock.
Options generally have an expiration date. Your original option grant contract document should have outlined when your options would ultimately expire. I have not seen options that last forever.
Moreover, employee stock options in particular are also likely to have other conditions that limit exercise after you quit or are terminated; for instance, one grant I've seen permits exercise of vested options only within 30 days of leaving, and unexercised options would be cancelled after that period.
An option is simply a right to buy a stock, typically at a preset price, and typically it is disabled until some point in the future. This is granted as an incentive to stay with the company.
When an option vests, that means it is now activated. Now you can buy the stock, if you want to, at that preset price. You don't have to. That option continues to exist for some time, but eventually terminates. One way it terminates is if you leave the company.
Typically when someone exercises employee options, they buy and immediately sell the stocks. (i.e. buy at the strike price, sell at market, presumably at a profit). This is so common that you don't even need to front the cash for the stocks - it is taken out of your sales proceeds automatically by the broker. You just select the "immediately resell" option (or rather, don't deselect it). It winds up being a nice cash bonus, which is exactly what it's intended to be.
The other choice there - rarely taken - is for the employee to buy the stock and hold it. In this case, the employee must pay for the stock with their own cash (the stocks are discounted, not free). That is the only case where options would result in something sellable in your portfolio.
There's an argument not to hold too much stock in the company where you work. Your livelihood is already there, and if that company fails, you could lose your job and your money!