What the problem is and is not
Maybe the notion of an object is primitive: it cannot be explained any further. A word is primitive if it is meaningful but not capable of being defined. Moreover, if the notion of objecthood cannot be explained in terms of something else, then, given that it is a genuine notion at all, then it is a primitive concept.
Maybe it would help to limit one's ambitions to the concept of physical objecthood. But it will not do to suggest that physical objecthood can be understood in terms of fundamental particles, such as quarks. Presumably, from a metaphysical point of view, the notion of a fundamental particle is if anything more mysterious than the notion of an ordinary physical body (a topic in philosophy of physics). Moreover, if particles are treated as bodies themselves, then to use them in explaining what bodies are is to offer a circular explanation.
The suggestion that physical objecthood can be understood in terms of fundamental particles belies a misunderstanding of the problem. It can be summed up, in a way to prevent this misunderstanding, as follows: is there any other category of being that can be used to explain what physical objects are?
Perhaps what objects are can be understood in terms of their own properties and relations. Bodies, for example, have properties and relations, and it seems that we can describe bodies only by mentioning their properties and relations. The only way we can talk about an apple, it seems, is by describing its properties, or how it is related to other things. For example, we can say the fact that it is an apple is a property of the object on the table; its redness is a property; its size and composition are properties; its being on the table is a relation; its being in the room, and being bigger than other apples, are also relations. So it appears that the only way we have of talking about the apple is to mention its properties and relations. Similarly with all other bodies--and, arguably, with minds and persons as well.
When philosophers raise the question, "What are objects anyway?" they want to know how objects are related to their properties and relations. We can leave aside relations now for simplicity, and ask simply: what is the relationship between objects and their properties? This is supposed to tell us better exactly what objects are.
Substance theory vs. bundle theory
The latter question can be explained further, as follows: "Are objects just the same as a collection of properties, or are objects not the same as, but different from their properties?" On first glance anyway, this appears to be a fairly strict dichotomy: either objects are the same as collections of properties, or they are not the same. So there are two theories of the nature of objects. The bundle theory, on the one hand, holds that objects are nothing more than collections of properties. The substance theory, on the other hand, holds that objects are something over and above their properties, namely substances. Philosophers have talked about both physical substances, or bodies, and mental substance, or minds. So the problem of substance arises in both the physical and the mental realms.
See substance theory and bundle theory for further explanation as well as some arguments for and against each theory. See also mind for discussion of bundle and substance theories of the mind.