Inside, don't leave food, birdseed, or pet food stored in bags. Use sealed plastic bins. Mouse traps or bait stations around the exterior perimeter are cheap insurance. For bugs, keeping the area around the building clear of tall grass, woodpiles, and debris removes the habitat that attracts them.
Our buildings are well-constructed with tight joints, which helps significantly. But a shed in a yard is always going to be more exposed than the inside of a house, so a little prevention goes a long way.
