Medium quests are, as you might expect, longer than short quests and are represented as a circle with 3 spikes sticking out in a triangular formation. It will also say medium under quest length on the left side quest summary.
The main difference here is that you will be given Firewood in your inventory which you may use in any cleared out room to camp (thanks to Zoot Marlowe pointing this out). While camping, you can eat to recover health/stress, but be careful that you leave yourself provisions in case you get a hunger event while wandering the dungeon. You can also spend points on camping skills to buff your characters, reduce stress, heal, remove diseases, and produce trinkets or supply items. Be aware that you can get ambushed in the night if you do not use a skill that prevents it.
All boss quests are of medium length.
Complete your medium quest and the achievement is yours.