So I'm trying to get into WC4, which I had a ball with as a teenager, but am dying really, really quickly. I was never that terrible at these sims, and do fine with WC3 (which is in DOSBox), but in 4 I seem to die from damage incredibly quickly - one second I'm at full health, the next I'm dead - and the enemy ships just zoom fast. It feels like the game is running on overdrive.

Is it possible that it's too fast on a modern system (i5 3470)? Or am I just a lot worse at this than I used to be?

I have the exact same impression as you have. The game seems too fast. I can't find anything on google though, so maybe it's all in our minds?

Maybe you played it on a computer that was 'too slow' for it back in the day?

Try using CTRL + F11 to slow DOSBox down.
It may do the trick.
You can raise the speed using CTRL + F12 if you think it's become too slow.

WC4 is a windows DVD game, actually.

Oh.. Didn't know that the GOG version was the DVD one.
I have the original game, it's both the DOS and the Windows 95 version, but on CDs.

After playing it a little bit more, it may also be that I'm just a bit rusty. Compared to Freespace 2 though, the gameplay seems a little bit less smooth. Maybe that's just because the game is older.

As a side note: Games seem to be far less forgiving back in the day. Nowadays you are expected to play through a game once and then move on to the next. Before, you picked a game you really liked and stuck with it until you got the hang of it. It required more skill.