Although Woolf does say that these two people are friends as well as enemies, she does not really mean that they are friends in the way we generally use the term. These are not people who really care about each other. Instead, they are more of allies -- they are allies and enemies at the same time.
You can call them "friends" because they need each other and they come to one another in time of need. The Duchess needs Oliver's money, Oliver wants the social status that comes with dealing with an aristocrat like the Duchess. So each has something the other needs and they are, in that sense, friends.
They are enemies at the same time because they are each (as Woolf says in the same sentence where she calls them friends) trying to cheat the other all the time. Each of them is really trying to exploit the other. This is how enemies treat one another -- just trying to use the other person to advance their own ends.