The amount that matters varies depending on the particular Zman, the time of year and location on this planet and on your specific concerns. For a simple example, the circumference of the Earth is 24,901 miles. Which means if you move ~ 0.2882 miles = ~ 1,521 feet (if I did the math correctly), your local sunrise and sunset times will shift by 1 second. At progressively higher latitudes the change happens over shorter distances.
However, while strictly speaking "as precise as possible" makes sense, keep in mind that:
- Until very recently, having a clock consistently accurate to within a second of "real time" was a luxury. And not that long ago, an impossibility. And if you go back a couple of hundred years, even a clock accurate to the minute was unusual and likely not found anywhere in the typical Shtetl, let alone on the Chazzan's Shtender.
- The most critical times in Halacha, start/end of Shabbos and Yom Tov (Melacha) and Yom Kippur (fasting) have a large built-in buffer, "Bein Hashmashos". This is a designated time that is added on each end. While in theory the actual change from one day to the next happens at a specific but unknown instant within Bein Hashmashos, in practice there is a general concept (at least for me) that this helps avoid "literal last second due to miscalculation or incorrect clock or inexact location" issues.
- MyZmanim and similar sources make it clear that you should not count on the times to the very last second. For my Shul (and, I am sure, many others), we round up or down all times "Lechumra" to the nearest minute.
Using Netz as an example. Let's say you have a stadium full of people who start Davening early and wait for the scoreboard to announce Netz to start Shemoneh Esrei. Logically that will be to the second for the center of the stadium. If it is far north, in theory one end of the stadium could be one or even two seconds off from the other end of the stadium. But Davening one second before Netz is not an invalid Davening. And if you are at all concerned, wait for the scoreboard notification, count two seconds in your head, and then start Shemoneh Esrei.
Go back a few hundred years, and the Zmanim still mattered. But Netz? You had a rough idea and timed your Davening approximately correctly and when you got through Shema you would wait until you actually saw the sun peek over the horizon. And Shkia? You had a rough idea, and would add 18 minutes before sunset to play it safe, but even then you didn't really know for sure, until you saw the sun setting. And Tzeis Hacochavim - it used to literally mean look outside at the stars - light pollution wasn't a problem.