Is there any Classical Mechanics system which needs to be described by a spinor? We need an ordinary number (scalar) to describe a harmonic oscillator, and a vector to describe, for example, a pendulum.
Is there any similarly simple system which we need to describe using a (two-component Weyl) spinor?
 A: Yes, there is a simple system which is efficiently described using 2-component spinors:  the dynamics of a classical spinning top. 
As Andrew Steane says, in his excellent undergrad introduction to spinors https://arxiv.org/abs/1312.3824
“It appears that [Felix] Klein originally designed the spinor to simplify the treatment of the classical spinning top in 1897. The more thorough understanding of spinors as mathematical objects is credited to Elie Cartan in 1913. They are closely related to Hamilton’s quaternions (about 1845).”
Since one definition of a spinor is its 720-degree identity under rotation, Hamilton’s quaternions, used to describe e.g. rotations in computer imagery and aeronautics, can also be considered as spinors of a relatively simple, but classical kind.   
For a more detailed discussion, see Chapter 5 of Laszlo Tisza’s MIT lectures on Applied Geometric Algebra ( https://ocw.mit.edu/resources/res-8-001-applied-geometric-algebra-spring-2009/lecture-notes-contents/Ch5.pdf ) 
“The two-component complex vectors are traditionally called spinors. We wish to show that they give rise to a wide range of applications. In fact we shall introduce the spinor concept as a natural answer to a problem that arises in the context of rotational motion.” 
Tisza’s chapter works through from triads to Cayley-Klein parameters and thence to 2-complex component spinors. Technically, his spinors would be non-relativistic Pauli spinors rather than Weyl spinors as they transform under SU(2) rather than SL(2,C). 
The same topic (no pun intended) is touched on by David Hestenes in his New Foundations for Classical Mechanics (2nd ed. pp. 466+) and, in terms of Cayley-Klein parameters only, by Springborn in https://arxiv.org/pdf/math/0007206.pdf (Goldstein, as advised by Alex Nelson above, mentions spinors in his 2nd edn (1980); but not in his 3rd edn (2002), nor, according to Hestenes in the 1st edn 1950 or 1951).
Late addition: David Hestenes paper Vectors, Spinors, and Complex Numbers in Classical and Quantum Physics addresses the OP's question more directly. 
"A wider use of spinors in classical mechanics ought to dispel the pervasive mistaken impression that spinors are a special feature of quantum mechanics."
But it should be borne in mind that Hestenes' definition of spinor rests on an interpretation of the i in the usual complex plane as a bivector, enabling him to label the complex plane as the "spinor i-plane". This interpretation is elaborated in 'Section 2-2. The Algebra of a Euclidean Plane' in his New Foundations for Classical Mechanics (op. cit.). 
Later addition: For those interested in the origin of spinors, although this isn’t strictly of a mechanical nature, one might take a look at a couple of papers by Jerzy Kocik, in which he argues that Euclid’s parametrisation of Pythagorean triples is (by a couple of millenia!) "the earliest appearance of the concept of spinors” https://arxiv.org/abs/1201.4418 and https://arxiv.org/pdf/1909.06994.pdf 
A: I'd be remiss if I didn't point out the "belt trick."  It's a physical system whose primary characteristics are that of a spinor.  Wikipedia's spinor page also has some rediculous examples with a sphere rotating while tied in place with belts.
This is actually a very important aspect in martial arts when dealing with joint locks.  Your feet are typically anchored to the ground.  If someone wants to rotate around you, and you can move in a simply connected way (such as that of spinors), they cant force you to give up your footing.  An extreme example of this is a Balinese Candle Dance.  In that dance (or any of the related ones), one spins a candle in a spinor like way, demonstrating the simply conected behavior.  One starts at one's feet, then drops to the knees, then sits down, and eventually does it with one shoulder on the ground, moving the rooted point closer and closer to the candle (decreasing how many joints you have available to accomplish the motion)
A: I'm not sure how to translate this, but quaternions can be used as a simple way to represent classical elliptical orbits.
I did a little bit of graphics to demonstrate this. You can hold down the right mouse button and move the mouse to see different perspectives. Click on each image to see the next.
After each quaternion single multiplication, you get the location in space and the time advance or delay when the orbiting entity is at that location.
