Theorems similar to Baire Category theorem

A variant of Baire Category Theorem states that if a complete metric space is a countable union of closed sets then at least one of the sets has a non-empty interior.

In my study I have a Banach space which is a countable union of closed sets $F_n$, with the property that for every $x \in F_n$, if $\lambda \in \Bbb{K}$ with $| \lambda| \geq 1$ then $\lambda x \in F_n$. I would need to find a set out of these which contains a circle(not necessarily a disk) around the origin(not necessarily centered in the origin), but containing the origin in its interior. I know this is almost impossible using only these conditions, but maybe I can find more properties of the sets $F_n$ before leaving this lead.

My question is like this:

Are there any theorems similar to Baire Category theorem which can prove something like this? Even if there is no theorem that directly solves the problem above, I am interested in any exotic variant of Baire Category theorem, in which maybe the sets $F_n$ have additional properties.

References are welcome. Thank you.

Not even true for $\mathbb{R}^n$ for $n>1$. Let $v$ be a non-zero vector with rational co-ordinates, and let $F_v = \{w: |w \cdot v| \geq |v|^2\}$. Let $F_0 = \{0\}$.
Then $\{F_v\}$ covers $\mathbb{R}^n$, but there is no sphere containing $0$ in any $F_v$, since no vectors perpendicular to $v$ are in $F_v$.
If you have circle in $F_n$ that contains $0$ in the interior, you can find a circle in $F_n$ which is centered on $0$.
Obviously, for $F$ to contain a circle around $0$, you need, at minimum, for each $v$, a $\lambda > 0$ such that $\lambda v \in F$. That's not sufficient, however.