I've taken star trails in my backyard, just to show my workshop students what not to do for star trails, but also to show them that they also shouldn't get too hung up on having everything totally dark to shoot star trails. For instance, I shot the above shot during a full moon.
More of the issue with doing star trails in cities is the coloration of the sky, which always looks kinda orange and horrible.
So I always drive at least 45 minutes out of the city, and most of the time, more than this to take photos of the night sky. We're lucky here in Los Angeles that although it's an enormous city, we can go to the mountains, desert or coast relatively quickly and take star trails or night photos. I can go to Vasquez Rocks which is 40 minutes away and have great night photos. Or go to Mt. Pinos, 1:15 away, and it's actually dark enough that I can get Milky Way shots. And with Milky Way shots, you definitely need dark skies, far more so than with star trails shots.