The reason why a function cannot be nested like a program is that it gets stored in the repository which is available to the entire run-unit and not just to one compilation unit.
Although you cannot use global variables within a function you can use external variables in order to share data other than the parameters which are passed to the function.
Example:
$set preservecase case repository(update ON)
id division.
function-id. myfunction.
working-storage section.
01 myexternal-data pic x(5) external.
linkage section.
01 myparam pic x(5).
01 myresult pic x(5).
procedure division using myparam returning myresult.
move myparam to myresult
move all "2" to myexternal-data
goback.
end function myfunction.
id division.
program-id. mytest.
environment division.
repository.
function MYFUNCTION.
data division.
working-storage section.
01 myparameter pic x(5) value "12345".
01 myresult pic x(5) value spaces.
01 myexternal-data pic x(5) external.
procedure division.
move all "1" to myexternal-data
move function myfunction(myparameter) to myresult
display myresult
goback.
end program mytest.