Can some mathematical objects exist without set theory? I'm a beginner in undergraduate mathematics, and am studying subjects like real analysis and abstract algebra. It seems that most mathematical objects hinge on set theory. A group is a set equipped with special properties, so is a ring, a field, and a vector space; a function is a subset of $ \mathsf{A} \times \mathsf{B} $, and a relation is a subset of $ \mathsf{A} \times \mathsf{A} $, and so on.

But without set theory, will these objects be ill-defined? I've spent a month learning about the construction of $ \mathbb{R} $, as well as proving things like $ \ a + 0 =a \ $ by studying Dedekind Cuts. Even a cut requires the notion of a set. I've realised that the intuition I had from childhood about what real numbers are is sloppy; I think I know how they work and what they are, but if you push me to explain what they are rigorously, I'm unable to do so. So if mathematical objects cannot be explained in terms of sets, they are at best hazy things with no clear definition.

But how did mathematicians come to realise that by studying special types of sets with special properties, wonderful things will emerge? How did they also avoid studying arbitrary boring sets?

Finally, I've read a few accounts about groups being highly geometric objects, with many exquisite "creatures" waiting to be studied, but what is geometric about a set with special properties? Likewise, I came into real analysis thinking that I can finally prove why $ \ a+0 = a \ $. Indeed I can, but not without a cost; I've been rather self-conscious about obvious things I once took for granted if I don't have a definition of them in terms of sets. 

Edit: My question differs from Zev's link because I want to know if mathematical objects hinge on sets (in other words, they exist outside of set theory), not whether there are mathematical objects which cannot be described by set theory.
Ultimately your question, as expressed in your Edit, is a philosophical one, not a mathematical one. If you're a formalist, then no, the objects of consideration have no independent existence, and, visualize what we may, only the formal systems, marks on paper and screens, really exist. If you're a Platonist, however, mathematical objects actually exist in the same sense that tables and chairs do, and axiom systems merely characterize classes of such abstract entities.

It's a question of which comes first, the theory or the (alleged) objects: do the axioms of, say, group theory "bring groups into existence" (existence at least in the minds of mathematicians)? or do they and did they exist independently, and the group axioms merely capture group-ness? Were groups created or discovered? 

There's no theorem that answers these questions, and there's no "right" answer; it's more a matter of disposition and belief.
According to Wikipedia, Alfred Tarski came up with a set of axioms for a substantial fragment of Euclidean Geometry.  The axioms require no set theory.  The axioms are first-order, stand as precise, and have gotten explored in an automated theorem proving context.

So, yes, mathematical objects can exist without set theory.

A specific function or a specific relation could just be something which satisfies some axioms.

For instance, if we have the axioms 


C(x, C(y, x)) 


and 


C(C(x, C(y, z)), C(C(x, y), C(x, z)))


A specific function C could be anything which makes the above two formulas true.  The same can hold for a specific relation.