Concept of set without concept of member? Are there any theory of objects, close to set theory, but where definition does not include the concept of set member and the concept of belongness?
Only such operations as intersection, union and so on?
This theory can be build like vector space theory. For example, empty set can be defined as object, which gives itself if intersecting with any other object.
And so on.
Is it possible to build interesting and apparently complete theory this way?
 A: There are several theories that capture part of the structure of sets without using elements.  If you just want the operations of intersection and union, you could work with lattices, or, to capture more of the properties of those set-theoretic operations, distributive lattices.  If you also want to talk about complements, then Boolean algebras become relevant.  There are lots of variations, depending on what operations you want to include and which properties of those operations you want to assume.  For example, there are lots of lattice-theoretic axioms weaker than distributivity. And Heyting algebras are like Boolean algebras except they have only those properties of set-theoretic operations that are constructively valid.  (Although I've listed several options, they are only the tip of the iceberg; I hope it's a big enough tip to be useful.)
A: Intersection and union generally refer to sets. The theory of locales and frames is like topology without points. They use a Heyting Algebra. See here.
Also you might read about lattices.
In these structures the operations are usually referred to as meet and join. If the structure has a least element that element will behave like the empty set.
A: The keyword you are looking for is structural set theory. The opposing notion — focused on membership — is material set theory 
There fairly are well-developed structural set theories. For example, the elementary theory of the category of sets — ETCS for short — is a good set-theoretic universe to do mathematics in.
There are also various approaches via type theory, but AFAIK those tend to be more oriented towards the needs of formal logic and computability theory.

In both cases, one can devise a suitable formal language so that doing mathematics in a structural set theory still resembles how mathematics looks in a material set theory; one just has a different interpretation of what the symbols mean.
For example, category theory teaches us that the notion of a morphism $U \to X$ is a very good substitute for the notion of an element of $X$, and this works especially well in the type of category described by ETCS, so great swaths of mathematical ideas can be adapted to structural set theory simply by replacing "element of $X$" with "morphism with codomain $X$".
This means one can study structural set theories in a relatively familiar manner, rather than having to give up everything one has learned about mathematics and adopt a new and alien language.
The new and alien language is still there, of course, to be used when you're comfortable with it. For example, a lot of category theory can be usefully depicted with diagrams of vertices and arrows, and there even exist formal diagrammatic languages!
