Inspired by this question (and a comment underneath it), I thought we should have a solid answer on the site for this question:
Can a character at 0 HP take damage?
Yes. This is implied by the rules on death saving throws (PHB p.197), which state in part (emphasis added):
If you take any damage while you have 0 hit points, you suffer a death saving throw failure.
Note that damage while you are at 0 hit points doesn't reduce your hit points, though:
A creature's current hit points (usually just called hit points) can be any number from the creature's hit point maximum down to 0. (PHB p.196)
Hit point mechanics change when a character reaches 0 HP. The character can still be hit with attacks and other damaging effects, but they don't take additional points of HP damage. Instead there are two variations:
1. Failed death saves
Instead of taking damage, a character at 0 HP that is hit with an attack immediately fails a death save (two death saves on a critical hit). This brings a character closer to death, but does not change their HP from 0. Notably, these effects are similar to the results of actual death save rolls: less than 10 causes a single failure, and a 1 causes two failures.
People would probably not refer to a failed roll as damage in quite the same sense that they use damage when referring to losing HP, but this is the effect attacks which would cause damage to a >0 HP opponent have on an opponent at 0 HP.
2. Massive damage (specifically for a target at 0 HP)
Massive damage also does not deal individual points' worth of damage but instead simply causes death, with no intervening steps. This mechanic is mathematically identical to massive damage at >0 HP, but is marked out as a specific case in the PHB.