I am having a hard time understanding the holographic geometry of a CFT on a circle. Say we have a CFT on a circle of perimeter $$L$$. And it extends in the interior to a gravitational solution, say a BTZ black hole or AdS. Consider the case of constant time. On one side, the perimeter of the circle is finite $$=L$$ and on the other side the interior geometry has a radius of infinite range $$r\in[0,\infty)$$. Can someone explain how is this possible?

What you're seeing is that if we limit ourselves to the simplest definition of "boundary", then AdS doesn't have one. Taking $$z \to 0$$ in Poincare co-ordinates $$$$ds^2 = \frac{R^2}{z^2} \left [ dz^2 + \eta_{\mu\nu} dx^\mu dx^\nu \right ]$$$$ for instance doesn't give the flat metric until we also strip off an overall factor that would diverge. Therefore what AdS does have is a conformal boundary. The gravitational solutions in it determine a CFT on an equivalence class of metrics related by a Weyl transformation.
This can be seen well in the notes https://arxiv.org/abs/1608.04948 by Penedones. Starting from Euclidean $$AdS_{d + 1}$$ defined as the hyperboloid $$$$-X_0^2 + X_1^2 + \dots + X_{d + 1}^2 = -R^2,$$$$ he discusses different ways to parameterize it. The metrics obtained this way just differ by co-ordinate choices from the bulk perspective. But looking at the conformal boundary, you can get flat space, cylinders and spheres to name a few. We can see that this is inevitable by going to asymptotically large $$X_i$$ where the hyperboloid becomes the null cone $$$$-P_0^2 + P_1^2 + \dots + P_{d + 1}^2 = 0$$$$ where the conformal group acts linearly. Nulity is only one constraint in a $$d + 2$$ dimensional space so, to describe $$CFT_d$$, we also have to choose a section of this cone. There are many sources including https://arxiv.org/abs/1107.3554 which discuss this construction not in a holographic context. But they are implicitly focusing on CFTs in $$\mathbb{R}^d$$ so they choose $$$$(P_+, P_-, P_\mu) = (1, x^2, x^\mu).$$$$ This is called the Poincare section which matches up nicely with the choice to use Poincare co-ordinates in AdS. But a compact boundary, such as the one you're describing, will require a different choice which is the one for global AdS.
The proper answer is that since AdS spacetime has infinite volume, we have introduce a cut-off to keep track of infrared divergences. This is called holographic renormalization. The boundary of AdS is taken to be at $$r = \epsilon$$. The CFT then lives on a circle of radius $$\epsilon$$. Calculations on both sides are performed with this cut-off present. When the cutoff is removed by taking $$\epsilon \to \infty$$, quantities on both sides often diverge. If everything has been defined and calculated correctly, the divergences in AdS precisely match those on the CFT side as does the finite part.
Interestingly the limit $$\epsilon \to \infty$$ has a very different interpretation on both sides. On the AdS side, it is an infrared limit (clearly!) whereas on the CFT side this is a UV limit. See this for details.