If you only need one relay to be on at a time, then things get much simpler. Arrange them in a 18x18 matrix.
SQRT 300 is < 18 so you just need to drive one of 18 lines HIGH, and one of 18 LOW.
3 shift registers for each of the HIGH and LOW drivers.
You'll still need approriate power handling devices , but if there are 'high-side drivers like the low-side TPIC6B595 you don't even need these.
you will need freewheel diodes, of course. 1N4148's are cheap
Allan