Brotato

Turrets can be purchased from the shop in game for materials they will appear in the shop as a turret to buy or can come with an item that can proc a turret spawn (Upon tree kill spawn a turret), simply have 5 of these on your map at once and the achievement will be yours. This achievement unlocks the Engineer class

Edit: Pocket factory (Turrets spawn from killing trees) DOES NOT count towards the cheevo therefore you need to purchase 5 different turrets from the store

Despite the description, the achievement isn't ONLY for actual turrets - it also applies to (at least some) other permanent structures as well. I got the achievement with no "turrets", but five gardens, all of which were created by the Pruner weapon, which creates a food-spawning garden on the map.

I was using the Pacifist class (unlocked by collecting 10000 materials, which will come naturally over a few runs), and I'd recommend that for this achievement - the Pruner is a starter weapon option for that class, and once you've got a weapon, additional copies of that weapon appear more frequently in the shop. After you equip a fifth Pruner, the achievement should pop at the beginning of the next round. I didn't specifically note which round I got the achievement on, but I believe I had it by round 10 or 11.

Because I got the achievement with five gardens, I can't confirm with certainty whether multiple structure types work (for example, three gardens, a Tyler, and a Medical Turret), although my assumption is that they would.