To earn this achievement, you must complete 4 sidequests (blue !)
- Brewery Upgrade (Base 1)
- Hepcat Upgrade (Base 2?)
- Boat Upgrade (Base 3?)
- Theater Upgrade (Base 4)
These optional quests are available from their respective bases and entail collecting 5 of some item.
Since you need to have the bases unlocked to get the missions, you need to progress in the story.
According to PhillyCh3zSt3ak, the 4th one becomes available after the quest "Getting the Band Back Together".
Most if not all are started by examining some pile of item in the base, which will cue Floyd to tell you to collect more of them.
Edit: Hepcat Upgrade has been known to bug out and I haven't found a clear fix online.