When I activate hard mode, will my world be destroyed?

Solution 1:

Upon defeating the Wall of Flesh a Corruption/Crimson and Hallow biome will generate in a large V, with each "edge" being either the evil or Hallow. The Terraria wiki states that

The Underground Corruption and Underground Crimson primarily replace Stone Blocks, Ice Blocks, Sand and Silt Blocks on their path with their respective evil versions immediately upon conversion.

The position in the world that the new biomes will generate is dependant on the seed.

Your chests and builds, assuming they are not made of those blocks, are safe.

In terms of spreading both new biomes will begin to spread much faster and more aggressively so could reach your spawn and base given enough time. Check out this article for making sure your world isn't overrun by new biomes.