Do emblems count as permanents on the battlefield?

Emblems are not permanents. According to CR 114.1, emblems are objects that exist in the command zone and have a characteristic called an ability. They are not cards and they are not placed on the battlefield. Permanents, by definition under CR 110.1, are cards or tokens that exist on the battlefield — emblems satisfy neither condition.

Because emblems live in the command zone (CR 114.2), they cannot be targeted, destroyed, exiled, or otherwise interacted with by spells or abilities that affect permanents. Effects that say 'destroy all permanents' or 'exile target permanent' have no effect on emblems whatsoever.

Emblems also have no card type (they are not creatures, enchantments, artifacts, etc.), so they cannot be counted or affected by type-based effects. They simply provide a continuous or triggered ability to their controller for the rest of the game.

Concrete example: If you activate Liliana of the Veil's ultimate and she gives you an emblem that reads 'Whenever you cast a spell, target opponent sacrifices a creature,' and your opponent then casts Merciless Eviction choosing 'enchantments,' the emblem is completely unaffected — it is not an enchantment, not a permanent, and not on the battlefield.

In summary: emblems are unique game objects in the command zone (CR 114.1–114.2) that grant abilities but are entirely separate from the permanent classification defined in CR 110.1.