Put the numbers 1 through 25 in A1:A25. Put this in B1 and copy down
to B25:
=RAND()
Select any cell in the A1:B25 and choose Data/Sort. Sort on column B
(ascending or descending, doesn't matter) with no headers. If you
need to save the sequence copy A1:A25 to a different location.
Since RAND() will recalculate after the sort, you can get another
random sequence by choosing Data/Sort again.
Since RandBetween() will produce duplicates within a data set, it
doesn't meet the OP's criterion of "simulating pulling the numbers 1
through 25 out of a hat"