What is the meaning of objects in a vector space?
Definition: A vector space is a nonempty set V of objects, called vectors, on which are defined two operations, called addition and multiplication by scalars, subject to the ten axioms...
Can entries of a vector be anything other than real or complex numbers?
A vector is an object. It does not need to have "entries"; this is a property that $\mathbb{R}^n$ has, but not most other vector spaces.
For example, the set of all real polynomials in the variable $t$, denoted $P(t)$, is a vector space. There are no entries; a vector looks like $1+2t-3t^5$.
Absolutely. Anything that can be "added" in a sensible way can be considered a vector. For example polynomials, and in particular, functions. In quantum mechanics this difference is almost ignored entirely; then functions are commonly referred to as "vectors".
Perhaps it would be clearer if the definition instead read:
Definition: A vector space is a nonempty set V, whose elements we will call vectors, on which are defined two operations, called addition and multiplication by scalars, subject to the ten axioms...
As pointed out in the other answers, the vector space can be absolutely any set on which you can define the required operations.
The vectors in a vector space are objects. These objects can be many different things. They can be ordered pairs with elements from a field $F$, sequences with elements from a field $F$, $m$ x $n$ matrices with entries from a field $F$, the set of all function from $S$ to $F$ denoted by $f(S,F)$ where $S$ is any non-empty set and $F$ any field, the set of all polynomials with coefficients from a field $F$ where $f(x)=a_nx^n+a_{n-1}x^{n-1}+\cdots+a_1x+a_0$ and $n$ is a non-negative integer and each $a_k$, called the coefficient of $x^k$, is in $F$. As you can see, there are many vectors that we can study in relation to vector spaces. In fact, everything I have just stated is a vector space with the operations of addition and scalar multiplication.
So you have this definition:
Definition: A vector space is a nonempty set V of objects, called vectors, on which are defined two operations, called addition and multiplication by scalars, subject to the ten axioms...
The axioms guarantee that $V$ under addition is an abelian (i.e., commutative) group. The set of scalars has to be a field. This means that among scalars the usual algebraic rules valid in ${\mathbb Q}$, or ${\mathbb R}$, or ${\mathbb C}$ apply. But there are other examples of fields, as explained in algebra or number theory courses.
Now to the question of "entries". In building a general theory of vector spaces the notion of dimension comes up. When a vector space $V$ over some scalar field $F$ has dimension $n\in{\mathbb N}$ then for all practical purposes (i.e., after choosing a basis) $V$ can be viewed as the set of all $n$-tuples $(\xi_1,\xi_2,\ldots,\xi_n)$ from elements of $F$. The $\xi_k\in F$ are the "entries" you spoke about. It is true that for vector spaces connected with physical or technical situations the scalar field $F$ tends to be $={\mathbb R}$ or $={\mathbb C}$.
