For Cheap: make it like a root cellar. Dirt is the best insulator and it will stay dark and damp. Also, your cooling costs are much less. The best cooling room I ever saw was in a basement corner of an old 2 story barn with rock walls.
Please don’t try to make it airtight unless you’re going to invest in a thermostat controlled electromagnetic door and/or an alarm system in case the power goes off or your cooling unit fails. Or unless someone can monitor it at least every hour. Without fresh air, the results can be fatal, very quickly. That’s not a chance worth taking IMO.
We used shade cloth around a pen under an open barn, with water coolers and fans. Not nearly as cool as you can get a sealed, insulated room, but you can grow hair with it, and it’s a whole lot safer than an unmonitored cool room.