How can I format page numbering in MS Word which is 01, 02, 03, 04, 05, etc?

Right-click a page number and select Toggle Field Codes from the context menu.

You should see something like { PAGE } or { PAGE \ * MERGEFORMAT }. Change it to

{ PAGE \# "00" } 

Press F9 to hide the field code and to update the result.

Source

1. Insert a standard page number
2. Press Alt+F9 to show field codes
3. The page number field will show up as either { PAGE } or { PAGE \* MERGEFORMAT } in a grey highlight
4. Replace this with { PAGE \# 0# }
5. Press Alt+F9 again to return to field values

For those who have MacOS, try Fn+Option+F9.