Junius Asks - "At what length does an antenna
begin to have directional characteristics?"

Different Antennas have Different Directional Characteristics :
[ All other things being perfect and technically correct :]

1/4 WL Vertical Wire Antenna - Omni {Full Circle}

1/2 WL Dipole Wire Antenna - The Broad Sides

1 1/2 WL Long Wire Antenna - The Far End

Note - Since this is expressed in Wave Lengths (WL)
these are the General Directional Characteristics at that
Specific Frequency (WL) Only and at other Frequencies
up-or-down the Spectrum then the Propagation Pattern
of the Antenna may vary slightly or very radically.

OBTW - Lack of any Directional Characteristics can be
viewed as a Directional Characteristic of sorts - of course ! )

so now let the real facts and personal opinions begin . . .
as to what is what -and- what is not - iane ~ RHF