The following will do it without (explicit) loops, but it is not a good idea to do it like this:
myArr1 = Array("a", "b", "c")
myArr2 = Array("1", "2", "3")
Range("A1:A3").Value = Application.Transpose(myArr1)
Range("B1:B3").Value = Application.Transpose(myArr2)
Range("C1:C3").FormulaR1C1 = "=RC[-2]&RC[-1]"
myArr3 = Application.Transpose(Range("C1:C3").Value)
Note: Because of the way they are created, myArr1 and myArr2 will be dimensioned 0 To 2, but myArr3 will be dimensioned 1 To 3.
A somewhat simpler method that doesn't require loops would be:
myArr1 = Array("a", "b", "c")
myArr2 = Array("1", "2", "3")
Dim myArr3(0 To 2)
myArr3(0) = myArr1(0) & myArr2(0)
myArr3(1) = myArr1(1) & myArr2(1)
myArr3(2) = myArr1(2) & myArr2(2)