How can I make a square hole 4 in. deep in wood or metal?

For metal there are special toolholders for metalworking lathes which are round on the outside, square on the inside. I’d bore a round hole and press-fit one of these into it. Available at McMaster-Carr, they have a great catalog and website. I’ll help you find it if you are interested. I think Enco has the same thing in their catalog as well. Otherwise, maybe some metalworking hobbyist might have a broacher, I can’t think of any other way to get a square hole in thick metal like that.

-- If Stradivarius was alive today, he'd be using Gorilla Glue.

you can also drill it with a 1/4” drill bit, and use a 1/4” (or smaller) chisel to square it off since most of the material will already have been removed in the drilling operation – use some scrap piece above the hole, to help you align the chisel straight down, and on the square line. take small bites.

for metal, filing I think would be the way to go – there are small rectangular/trignale files that can reach in and are thin

-- When in doubt - There is no doubt - Go the safer route.