Here's a GUI way to do it: Paste them into two columns in a spreadsheet, copy all cells out, then use regular expressions to replace tabs with newlines.
line 1 from file1
line 1 from file2
line 2 from file1
line 2 from file2
...etc
Using awk can be useful if you want to add some extra formatting to the output, for example if you want to label each line based on which file it comes from:
Note (of interest to me) this needs a little more work to get the ordering right if instead of static files you use the output of commands that may run slower or faster than one another. In that case you need to add/sort/remove another tag in addition to the line numbers: