On Fleek

It helps to use the safe in Sanctuary to store items for this so you don't have to carry them around all the time. I.e. it doesn't matter what level or how good the items are - only their colour.

Note the 4th weapon slot only becomes available after completing the story mission "Beneath the Meridian", and the artifact slot only after "Cold as the Grave" (which is much later).  You therefore can't get this achievement until after that point...