Do crystals form like that naturally?

Yes, in mineral-rich water over millennia.

The crystals kept growing till 1985, when the cave was discovered and and drained by the miners. Even though the crystals resemble icicles and suggest it should be bloody freezing in there, the temperature actually reaches 112 degrees F, and the humidity measures up to 90-100%. Such conditions make it impossible to enter without a protective suit and backpacks filled with ice-cooled air. Naica lies on an ancient fault and there is an underground magma chamber below the cave. The magma heated the ground water and it became saturated with minerals, including large quantities of gypsum. The hollow space of the cave was filled with this mineral-rich hot water and remained filled for about 500,000 years. During this time, the temperature of the water remained very stable at over 50 °C (122 °F). This allowed crystals to form and grow to immense sizes.

Millennia are so long compared with a human lifetime but so short compared with the planet's age.