Indeed, it's "just close enough." Pi is a irrational number, and will continue indefinitely without repeating (which is how science fiction shows like "Star Trek" tend to mess up computers or robots that are running amok, by asking them to express it to the last digit). 3.14 is used to represent Pi Day as they are the most significant digits of Pi.
Some are calling March 14th, 2015 at 9:26:53 AM the "Real Pi Day." These numbers (3/14/15, 9:26:53) correspond with the first 10 digits of pi (3.141592653). If you were to round that last digit, though, it would really be 3.141592654 as the next digit is 5, meaning to 11 digits the value of pi is 3.1415926535. Indeed, that's all "very picky."
As an irrational and transcendental number, pi has no complete representation. However, with the use of computers, pi has been calculated to over 1 trillion digits past the decimal point. You can actually view 1 million digits of pi here.
Pi is the ratio of the circumference of a circle (distance around its edge) to its diameter (width at its widest point). Pi Day is celebrated by math enthusiasts around the world on March 14th. One common thing to do on Pi Day is to eat, or even bake, a pie. Pi-designed pies are especially appreciated by scientists and mathematicians on Pi Day.
When did this faux holiday commence, you might ask? Physicist Larry Shaw created Pi Day in 1988, and it was first celebrated at the San Francisco Exploratorium, where Shaw worked. On that day (March 14, 1988, obviously), Exploratorium staff and the public marched around one of its circular spaces. The group then followed it up by eating fruit pies.
You can see an image of Larry Shaw with some pies at the Exploratorium, above.
The Exploratorium continues to celebrate Pi Day. This year, they will have a webcast on explo.tv (beginning at 1:00 p.m. PDT) and they will also celebrate in the virtual world Second Life.
There's another day, later in the year, also celebrated as an homage to pi. That day, Pi Approximation Day, is July 22. Much as 3.14, 22/7 is often used as a rough approximately for pi, thus the naming of July 22 as Pi Approximation Day.