It's actually amusing that, just few weeks ago, people here on HN were arguing on why are we trying to quantize gravity and that it makes more sense for it to be non-quantum.
And if gravity is continuous, could it store infinite information (for all the same reasons, having arbitrarily precise values) then?
Of course the field can be continuous, even if the weight is exactly some rational number. "Discrete steps" just means that the value differences in gravitational waves is also discrete. A bigger problem, where relativity clashes with QM, is stuff like singularity and superposition. We don't have an answer for that yet.