It is possible, except there is one region that contains
only one member.  Zero is the only whole number which is
not a coubnting number.

Draw the figure below. 


 

The small (red) circle will represent the set of counting numbers
The medium sized (green) circle will represent the set of whole numbers
The largest (blue) circle will represent the set of integers 
The square on the left will represent the set of rational numbers
The square on the right will represent the set of irrational numbers
The big rectangle, consisting of the two squares and all the circles
       will represent the set of real numbers

Inside the red circle write the counting numbers 1,2,3,4,5, or
any other five counting numbers you choose.

In the region outside the red circle but inside the green circle
write the number 0 (zero), which is the only whole number that is not a 
counting number.  Note that as I said above, it is impossible to put 5
different numbers in this region because there is only one such number,
zero.

In the region outside the green circle but inside the blue circle, 
write five negative integers, say -1, -2, -3, -4, -5 or any other five
negative integers you choose, for the negative integers are the
only integers which are not whole numbers. 

In the region outside the blue circle but inside the left square
write these 

, , , , 

or any combination of 5 of these:

1. common fractions, positive or negative, which do not reduce to integers.
2. terminating decimals, positive or negative, such as -2.876
3. repeating decimals, positive or negative, such as 7.483483483483...
   or 

In the square on the right, write these:

, , , , 


or any other irrational numbers you know of.  Note that the decimal
above  is NOT a repeating decimal, because
it does not repeat the same block of digits over and over as does the
decimal  or . 

I can't write all those numbers in the Venn diagram above, but you can
on your paper.  Just draw your figure big enough to write them all in
there in the right places.  Be sure to point out that there can only
be one number in the set of whole numbers that are not counting numbers.

Edwin