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You can do it. Downside is you lose useable area in the room. Rigid insulation board probably better than floppy loft stuff. You want to try and prevent water vapour getting into the frame/ onto the wall though, otherwise you will get condensation against the cold wall and hidden mould. Foil-backed insulation board is a thing intended to make vapour resistance easier. Tape the joints.
So...
Am I being crazy?
Cold rooms yeah. External wall.
What's stopping me from knocking a frame together, putting insulating material (loft stuff), putting sheet of plywood over frame. Painting.
Is that mad?