Why are stable fundamental particles fermions? Within the Standard Model, any particle decay eventually terminates at the same stable fundamental particles, i.e. u- and d-quarks, the electron and neutrinos (let's forget about neutrino oscillations for now), or the respective antiparticles. Those are fermions. Is there a (simple) answer why? Could one, in principle, make a consistent model where the stable fundamental particles are bosons?
 A: The standard model evolved to fit measurements and observations. The observations  have an axiomatic position in any model designed to fit the observations. 
Mathematics allows to pick different sets of axioms, making theorems into axioms and former axioms provable as theorems. In a similar way the postulates which tie a subset of mathematical relations to describe the measurements can be changed, and new , mathematically simpler postulates replace them, but the observational ones are still there to be proven from the postulates. In other words in a physics model there are always statements that are axiomatic, connected with observations. A final un-peeling of the onion , asking "why", will at the end hit a "because that is what is observed".
So the "why are fundamental particles fermions" hits on baryon number conservation as a validated hypothesis, and the observation that the proton has spin 1/2 as well as the electron has spin 1/2, and both are stable.
From these three observations/measurements  and conservation laws, the complexity of particle interactions, starting from scatterings of protons, electrons and photons, have disclosed the number of fundamental  particles in the table:

Conservation laws, also basic and axiomatic from observations, of momentum, energy, and angular momentum have led to this table , so that a mathematically consistent model, the standard model could fit and thus encapsulate the plethora of data.
Thus it is the two everyday stable particles of protons and electrons that are underlying the spin determinations of all the elementary particles in the table using consrvation laws. It so happens that it is only the photon that is a stable boson, but the "happens" is a physical observation.
In other words, thinkers did not sit and think " let us assume that most stable particles are fermions and see if this fits the data, if there exists a stable world". The mathematical model evolved to fit the data.

Could one, in principle, make a consistent model where the stable fundamental particles are bosons?

If you are asking about the world we live in, the answer is no: because the stability of matter as we know it depends a lot on the Pauli exclusion principle, also an observational postulate.
A: 
Within the Standard Model, any particle decay eventually terminates at the same stable fundamental particles, i.e. u- and d-quarks, the electron and neutrinos.

Not true - decay chains can also end in stable photons, and at temperatures high enough to create quark-gluon plasmas, they can also end in quarks or gluons.  In general, particles decay into lighter particles whose total decay products have the same total conserved quantum numbers of color charge, weak isospin, electric charge, weak hypercharge, baryon number, electron number, muon number, and tau number.  (Neutrino oscillations allow interactions that violate electron, muon, and tau number individually, but preserve their total "lepton number.")
A: Think it by the way that the stability of the properties of the subatomic particles in our surrounding allows us to find the same conditions around for life. A knife would be a knife in Africa as well as in Australia as well as in the Space station.
There are some moments which are important:


*

*If the energy exchange between the subatomic particles would be binary - means it would only exist photons of one energy content - the electrons in atoms would be only in two states, excited or not, and this would be a very poor world. So the bosons are a class of very important particles.

*Photons have both oscillating electric and magnetic dipole moments. But only two photons - of the same energy - can have a symbiotic state (photon bunching), bigger clusters (connected through their magnetic and electric dipole moments) are not possible.

*The above mentiond stability of our surrounding is a local property only. Near conglomeration of matter, much larger in relation to our sun, the gravitational potential transforms the accustomed world. Electrons or protons no more exist, only neutrons. Under higher gravitational potential neutrons no more exist, only gluon-quark plasma. In Black holes - to be speculative - perhaps even the photons are smelted in constituents.


What the science has done is the classification of the founded elementary particles into the Standard model and the division of this particles into matter constituents and exchange (of energy) constituents.
