Rigid monoidal abelian categories over a field $k$ with a fiber functor (i.e., a strong monoidal exact functor) to $K$-vector spaces for some extension $K$ of $k$ (the functor is assumed to exist, but not fixed) are are often called tannakian categories; a tannakian category with a fixed fiber functor to $k$-vector spaces is referred to as a *neutral* tannakian category. I will adhere to this terminology.

It was proved by Grothendieck, Saavedra-Rivano and Deligne that the 2-category of tannakian categories over a fixed field $k$ is equivalent to the 2-category of affine gerbes over $k$. With each affine gerbe $\Gamma$ you associate the category of vector bundles on $\Gamma$. Conversely, given a tannakian category $\mathcal A$, the objects of the corresponding gerbe over a $k$-algebra $A$ are fiber functors from $\mathcal A$ to finitely generated projective $A$-modules.

So, if $\Gamma$ is an affine gerbe, fiber functors from the category of vector spaces on $\Gamma$ to vector spaces on $k$ correspond to objects of $\Gamma(k)$; hence, to give an example of a tannakian category without fiber functors it is enough to give an example of an affine gerbe without a rational point. There are many such examples; for example, you can take a surjective homomorphism of algebraic groups $E \to G$, and a $G$-torsor $P$ over $k$ that does not lift to an $E$-torsor. The gerbe of liftings of $P$ to an $E$ torsor is an affine gerbe, and does not have any rational points.

As to examples of rigid monoidal abelian categories without fiber functors to any extension of $k$, the standard one is the category of $\mathbb Z/2\mathbb Z$-graded vector spaces.