First of all the question in the title of this thread cannot be answered without quoting the conventions used for the ##\gamma## matrices.
The general answer is that an otherwise unrestricted Dirac spinor describes a particle with spin 1/2 and an antiparticle with spin 1/2.
To see this, the most simple and consistent approach is quantum field theory, which is simple and consistent, because it's the natural language to discuss relativistic QT, which usually deals with situations, where the particle number is not conserved, because you consider scattering processes with energy exchanges in the order of the mass of the particles and larger, and thus with some probability in the reactions particles can be destroyed and new ones can be created.
To define what describes "particles" you need to consider free, i.e., non-interacting relativistic fields first. Poincare invariance then dictates how possible wave equations look like. Further the Poincare group tells you by analyzing its irreducible unitary representations that free fields obey the Klein-Gordon equation, which gives the "dispersion relation" between frequency and wave number, ##\omega=\sqrt{m^2+\vec{k}^2}## (in units, where ##c=\hbar=1##). Then also you have always also solutions with ##\omega=-\sqrt{m^2+\vec{k}^2}##. This defines the "modes" of the solution, i.e., plane-wave solutions. In the 1st-quantization formalism this are the momentum eigen states, and for free particles this implies the relation between energy and momentum, but this interpretation fails here, because of the negative-frequency modes.
Now help comes from field quantization (2nd-quantization formalism). All you have to do is to make the fields operator valued and decompose the fields in plane-wave solutions. To get only states of positive energy (and the energy spectrum should be bounded from below, because there should be a state of minimum energy, such that we have stable particles as observed in Nature), you have to simply write an annihilation operator in front of the positive-frequency modes and an creation operator in front of the negative-frequency modes. That's the Feynman-Stückelberg trick without the esoteric handwaving about "particles with negative energy moving backwards in time".
To the contrary this simple mathematical trick makes the theory not only having a stable ground state but also local, i.e., it let's you fulfill the microcausality condition, according to which local observables (energy denisty, momentum density, angular-momentum density) commute if their arguments are space-like separated, which guarantees the Poincare invariance of the physical quantities you can calculate using the theory (most importantly in HEP physics are the S-matrix elements which let you predict cross sections to be measured in the scattering experients at the large accelerator facilities like CERN).
Of course, in addition to momentum and energy there's also angular momentum, and as in non-relativistic QT you can have half-integer and integer angular-momentum representations. Then one of the fascinating miracles occurs: It turns out that in order to fulfill the microcausality condition and the existence of a stable ground state you must quantize half-integer-spin fields as fermionic and the integer-spin fields as bosonic quantum fields, and this is always confirmed by observation.
Now we have a clear particle interpretation for quantized free fields with the remarkable property that, just assuming microcausality (i.e., the consistence of the theory with the causality structure needed to make sense of any relativistic theory) and the existence of a ground state (stability of matter), you conclude that for every particle there should be also an anti-particle, which has the same mass as the particle and opposite charge. As a special case any kind of field also allows for "strictly neutral" particles, i.e., such particles with all kinds of charges being 0. Then the particles and antiparticles are identical (e.g., the photon as the quantum of the electromagnetic field, although there one needs to be a bit more careful, because it's a field with 0 invariant mass, where you have to analyze everything a bit different than with massive fields; e.g., you have a field of spin 1 but only 2 and not 3 polarization degrees of freedom; all this also follows from the analysis of the proper orthochronous Poincare group), and you know that necessarily half-integer-spin fields lead to fermionic and integer-spin fields to bosonic particles.
A Dirac field is not the case of a field with spin 1/2, which describes a particle (e.g., the electron) and its antiparticle (the positron), which are fermions. It has four components, because for a particle of spin 1/2 the spin component in ##z##-direction (or any other arbitrary direction, but the ##z##-direction is the standard choice in QT) takes two possible eigenvalues, ##\sigma_z=\pm 1/2##, and there's also the anti particle which has also spin 1/2 with also two possible ##\sigma_z##-eigenvalues. So you have 4 components.
