And why is an antenna resonant, because the voltage and current are in phase or because the reactances cancel out, or both ?
Reactance ($X$), resistance ($R$), and the phase difference between voltage and current ($\theta$) are related by:
$$ \tan \theta = { X \over R } $$
So if reactance is zero then $X/R$ is zero, so $\theta$ must be zero, meaning voltage and current are in phase.
In other words, "zero reactance" means "voltage and current are in phase". One does not cause the other: they are two ways of saying the same thing.
Likewise, resonance is defined as zero reactance. It's not "caused" by zero reactance or voltage and current being in phase: it simply is, by definition.
In other words, if you design an antenna with any one of these three objectives:
- current and voltage are in phase at the feedpoint,
- feedpoint reactance is zero, or
- antenna is resonant
you will find the other two are satisfied. There is no ordering or causality between any of them: they are each mathematically equivalent.
If you are concerned only about the feedpoint, asking what the inductive and capacitive reactances are which sum to zero reactance is like solving $x-x=0$ for $x$. It's not possible, and also it doesn't matter.
Besides, a dipole isn't an RLC circuit, so there is no inductance in henrys or capacitance in farads that can be found, although as Brian K1LI explains it is possible to approximate a dipole as an RLC circuit over a limited frequency range.
There is perhaps a slightly different question you could ask which does have an answer. A dipole, like any resonant system, has a Q factor. We know that a dipole must store some energy in its electromagnetic field, and some of this energy is lost to radiation and possibly other losses, like resistive losses in the conductor and ground. The Q factor relates to the ratio of these quantities.
One way to think of Q is as the ratio of reactive power to real power. The reactive power has current and voltage 90 degrees out of phase. The energy associated with this power doesn't go anywhere and doesn't do any real work: it just oscillates between the electric and magnetic fields, forever. The real power has voltage and current in phase, and does real work: mostly (ideally, entirely) radiation.
A good half-wave dipole will have a Q of about 10. Which means if the antenna is radiating 100 watts, there is about 1000 watts of reactive power. You could measure this with electric and magnetic field probes in the near field of the antenna. With the electromagnetic fields known you can calculate the Poynting vector. In the near field of the antenna you should expect to find the imaginary part of the Poynting vector is about 10 times that of the real part, meaning the electric and magnetic fields are almost but not quite 90 degrees out of phase. Of course the precise values will depend on just where you place the probes, and the construction and environment of the particular antenna.
While this is not exactly what you asked, it does seem closer to the understanding you're seeking. Unfortunately it does require thinking beyond just what's happening at the feedpoint, and instead considering what's happening in the electromagnetic fields in space around the antenna, and this requires some more complicated mathematics.