What does it mean when I hear giant gates and chains while mining?

Nothing.

What you're hearing are ambient noises, these are noises that Minecraft randomly plays sometimes. They are made to be creepy, but don't actually do anything other than sound weird. A quick search reveals this which meets your description quite perfectly, I think.

It indicates a change in the time of day, like day to night and vice versa. Source? I've played a lot in Survival mode and that really freaked me out when it came out a few updates ago (I think aquatic). I believe that they added it to help when you are down in the underground to know that time has changed aboveground.

Those sounds are 'cave ambience', and hearing them means you should be careful; you're in the dark, and monsters might be able to spawn. No 'thing' in particular is making the sound; Minecraft is actually giving you a helpful warning.

Cave ambience is not random. It does not indicate that any other event has occurred, but it does provide information about your current location. Minecraft plays ambient cave sounds play when a mood value reaches 100%, and then the mood resets to 0 and begins to increase again. The rate at which the mood value increases is based on an algorithm which considers various factors including depth and darkness, according to this. I haven't reviewed the code myself, but my personal experience supports this claim.

Since branch mining is typically done deep underground, cave ambience sounds tend to play very shortly after you enter a 'dark' area. Thus, they can serve as a quick warning when you've forgotten to light up a branch mining path sufficiently.

So, I always take cave ambience sounds as a reminder to be cautious, but if I unexpectedly hear them while I'm in my own mine, I know I have a lighting problem, which I'd better fix soon if I want to avoid unwelcome guests.