Well… Short answer, no. Long, who the hell knows. Gems are rocks, everyone being formed by some amount of heat and pressure. So, polymorphic sentient rocks (thank you, Ronaldo) have no possibility of existing ON EARTH. read more
While it may be cool to see colorful, weapon-wielding gems in real life, It is technically impossible for that to happen (Until we get Steven Universe Go, If you catch my drift ;)). To catch on why it is impossible, we need to know what gems are. read more
In conclusion the gems could be considered living things but many of the requirements need the use of shape shifting which excludes peridot and presumably several other era 2 gems so the era 1 gems(garnet, amethyst, pearl, lapis lazuli, jasper and the diamonds) are living creatures while era 2 gems like peridot are not. read more