Antenna conterpoise or ground plane. Confused about length requirements
If you are using radials, they should have a length of \$\lambda /4\$ (or an odd multiple of that). Consider the end of the radials as a high impedance (open) point that gets transformed by a \$\lambda /4\$-transformer to a low impedance (ground) point.
A balun is not necessary, since both the antenna and the cable are asymmetrical. In practice, however, the use of a common mode filter (coaxial core) may still be necessary.
You can try to improve the impedance matching by pointing the radials slightly downwards. In theory, radials that are perpendicular to the radiating element will result in an impedance of about 36\$\Omega \$. If the radials point straight down, the theoretical value is about 73\$\Omega \$.