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Assume character A is in the AOE of a damaging spell.

Character A is carrying a closed container inside his backpack, alongside several sheets of parchment.

Damaging spell description specifically states that items carried by a creature are not damaged even on a failed save. Unattended objects are damaged, though.

Inside said container there is a small creature. Creature has no problem living inside container for a long amount of time.

Does this creature take damage from the spell? If it does not, what mechanic prevents damage?

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The creature in the container does not take damage. From the rules on Areas of Effect:

If no unblocked straight line extends from the point of origin to a location within the area of effect, that location isn’t included in the spell’s area.

Since there's both a backpack and a container between the creature and the spell's point of origin, there's no unblocked straight line, and the creature won't be affected.

You've stated that objects are unaffected, but for other readers who might be thinking "Doesn't the container break?" - technically, no. Each spell states what it does or does not affect. For example, fireball:

Each creature in a 20-foot-radius sphere centered on that point must make a Dexterity saving throw.

[...]

It ignites flammable objects in the area that aren’t being worn or carried.

So a fireball will affect creatures within its area, and objects that aren't being worn or carried. The container is an object, but it's being carried, so it's fine.

Going through each and every spell to check whether they affect objects is left as an exercise for the reader, but very few spells affect objects, and even less affect objects being carried by a creature.

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  • \$\begingroup\$ Comments are not for extended discussion; this conversation has been moved to chat. \$\endgroup\$ – doppelgreener May 21 '17 at 15:28
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The DM must decide what "blocks" the spell.

As noted, the area of effect of a spell can be blocked by intervening objects. However, there is no absolute guidance regarding what sorts of objects and materials are substantial enough to block the spell. The DM must use his discretion.

I think that almost all DM's would rule that putting the creature in a tinfoil cup and setting it on your shoulder should not protect against a fireball - though per Miniman's answer, it does, for there would be no direct line to the creature itself. (It seems hard to imagine that a bit of tinfoil offers more protection than a suit of plate armor.) But most DMs would be fine with the idea of protecting the creature with a sturdy box, tucked deep in the backpack.

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