Some of the best strawberries I have had in recent years were raised off the ground in gro-bags. A wooden frame with netting between and a gro-bag plonked on top. Cut the gro-bag, plant runners. Worked really well and seemed easily refreshed.
IMO a better way of doing it would be to make a raised bed on stilts, set up a no-dig planter on top and grow strawberries in there. Painted white, it would reflect a lot of sun and the underflow of air would keep them cool.
Just read your post better, your steel drum could do with being painted white, as white as you can get. A newspaper lining would just rot down and make a good contact with the hot drum. What you want is to stop the drum heating in the first place. Insulate the outside with loft insulation/rockwool. It won't rot and will keep the outside of the drum a lot cooler in the summer and warmer in the winter.