How to transpose array of strings in Julia?

Why?

["a" "b"]' does not work because the ' operator actually computes the (lazy) adjoint of your matrix. Note that, as stated in the documentation, adjoint is recursive:

Base.adjoint — Function

adjoint(A)

Lazy adjoint (conjugate transposition) (also postfix '). Note that adjoint is applied recursively to elements.

This operation is intended for linear algebra usage - for general data manipulation see permutedims.

What happens in the case of [1 2] is that adjoint does not only flip elements across the diagonal; it also recursively calls itself on each element to conjugate it. Since conjugation is not defined for strings, this fails in the case of ["a" "b"].

How?

As suggested by the documentation, use permutedims for general data manipulation:

julia> permutedims(["a" "b"])
2×1 Array{String,2}:
 "a"
 "b"

Transposing is normally for linear algebra operations but in your case the easiest thing is just to drop the dimension:

julia> a = ["a" "b"]
1×2 Array{String,2}:
 "a"  "b"

julia> a[:]
2-element Array{String,1}:
 "a"
 "b"