I imagine a reactor operating at 1000C is built almost entirely on graphite and ceramics... But I have no idea how they plan to use it with water, as graphite doesn't behave well on those temperatures with water around. (Maybe there is a heat exchanger at some point that makes failures on the wet side safe.)

It certainly looks possible. You will probably end up needing some amount of metals on the hot portion, but they can be minimized to a very large extent. Cheap, on the other hand, is not a property I would guess from that description.