If you go Bullberry's website (Fred makes custom barrels for the Contender and Encore, as well as other stuff), very detailed information is posted showing that the .204 Ruger excels best at the 23inch length, and the 17HMR excels at the 21inch lnegth.
I know the .22 Long rifle excels at the 14 to 16 inch length, depending on the specific loading. By the same token the .22 Short is supposed to be at its best with about 11 inches of barrel, and the .22 WMR is at its best with around 19 inches of length.
I realize the rimfire rounds are all straight-wall cases, and the .17HMR and .204R are shoulder-ed rounds .....
The .22Hornet, .218 Bee and .22 Jet have minimum taper or shoulders, whether one would would call then straight or not.
Data I've come across about the .221 Fireball shot as a rifle show that that cartridge almost duplicates the standard .222 in possible velocities and bullet (weight preferences). I'd expect the Fireball to generate velocities within 100fps of the "duece," with bullets no heavier than 50-52 grains, on barrels no longer than 22inches ......
Twas me I'd pick a barrel length and taper that I liked and just run with it!
I've personally studied the same four rounds Brian asked about, to shoot the Fireball as a rifle, but the other rounds as "candidate cartridges" for reloading loads similar to rimfire velocities - just because that's a niche I wanted to scratch!
To determine what the best barrel lengths are for those four rounds, try to measure possible needed barrel length along with powder capacity in each casing and expansion ratio compared to the .22WMR and .17HMR.
T'was me I'd go with an 18in minimum length with the Hornet, K-Hornet or "Jet," and 20inches for the Bee and Fireball. Doubt I'd go longer than 20inches for the Hornets or Jet, or 22inches for the Bee or Fireball.
I'd also stay away from thin whippy barrels, and opt for standard tapered barrels or even heavy tapers if the extra weight made it easier to hold the rifle while aiming esp off-hand.
Either way, in a HandiR all these rounds will make short compact singleshots, and the added weight from a heavy barrel might be an advantage! Of course opinions are likely to vary!
Re-Read LaOtto's post. On the Bullberry website the owner took .204R and .17HMR Encore barrels that were 24inches long, chronographed them, then chopped an inch off the length and chronograohed each of them again.
Did this over and over agin down to 18in for the .204R and down to 14in for the .17HMR. It was easy to see where the "margin of diminishing marginal returns" kicked in with each cartridge, and of course Fred used one "standard" load for each "test."
When determining what barrel length is "ideal" is a very personal choice, hence my comment about choosing a favorite length and running with it!
If I was going for light weight and compactness, I'd still not take anything in a barrel less than 18inches and a std contour taper, but if I was more interested in the best accuracy possible in any of those rounds, the "minimum" I'd want would be chambering each of those cartridges in a 20inch HB barrel. But that's me.