Is it prohibitively larger? And is the size a fundamental constraint of the technology, or is it possible to reduce the size?

The size is a fundamental constraint of optical technologies, because it is related to the wavelength of light, which is much bigger than the sizes of semiconductor devices.

This is why modern semiconductor devices no longer use lithography with visible light or even with near ultraviolet, but they must use extreme ultraviolet.

The advantage of such optical devices is speed and low power consumption in the optical device itself (ignoring the power consumption of lasers, which might be shared by many devices).

Such memories have special purposes in various instruments, they are not suitable as computer memories.

Previous reply is correct.

To give a feeling: micro-ribg resonators are anywhere between 10 to 40 micrometer in diameter. You also need a bunch of other waveguides. The process in the paper uses silicon waveguides, with 400nm width if I'm not wrong. So any optical feature unfortunately isn't going down as much as CMOS technology.

Fun fact: the photolithography has the same limitations. They use all kinds of tricks (different optical affects to shrink the features) but fundamentally limited by the wavelength used. This is why we are seeing a push to a lower and lower wavelengths by ASML. That + multiple patterning helps to scale CMOS down.