Further to Grzesio's comments, one should note that Artemis was more of a course corrected 5-inch rocket, rather than a missile in the modern sense.
In "British Destroyers and Frigates", Friedman mentions that the guidance range of a beam-riding missile is much greater than a semi-active homing weapon due to the limited power of the illuminator. I find this a little confusing, as the receiving SARH missile is closer to the target than the receiving director in a B-R system. Perhaps it relates to the size and sensitivity of the missiles receiver, and choice of frequency?
There is also the issue of changes in signal strength as the SARH missile closes on the target - IIRC this was a problem in the development of the BAT active homing anti-ship weapon.
RP1
It's not so much the power of the illumination radar but rather the dispersion of the beam. As the distance grows longer, the beam gets wider as does the reflected energy from the target. This means the missile has a harder time distinguishing where the target actually is within the return.
Many early SAM's like Nike and Talos got around this issue by incorporating both systems into the missile's guidance. It also permitted the missile to fly a very fuel efficient ballistic trajectory to the area of the target using beam riding. Once close to the target, it was illuminated and the missile could home accurately on that target.
The problem with both missiles was the size of the tracking and illumination radars had to be huge to get sufficient accuracy to guide the missile. The other issue was developing a guidance computer that could align the radars accurately and keep them centered on the target.
Beam riders that fly directly to the target are very energy inefficient in terms of flight to the target. Semi-active systems using a collision course intercept are better.
With Artemis, the guidance system was particularly crude and what the missile really needed was a control system that allowed for a reasonable level of precision. This would have greatly complicated the design however. First it would require the missile be gyro-stabilized to remain steady in flight and have a zero reference plain as it flew. Next, control surfaces would be necessary in addition to stabilizing ones to make the flight smooth and turns accurate.
My guess would be that you could do it in an 8" rocket, possibly a 5," but nothing smaller.
The original Artemis design tried to cram everything into a new head that fitted on the front of the existing missile, and that just wasn't going to work. You also really need a proximity fuze to give a good Pk, something that had yet to enter service.