No, the cache is a few GB large for most usual context sizes. It depends on model architecture, but if you take Gemma 4 31B at 256K context length, it takes 11.6GB of cache
note: I picked the values from a blog and they may be innacurate, but in pretty much all model the KV cache is very large, it's probably even larger in Claude.
Yesterday I was playing around with Gemma4 26B A4B with a 3 bit quant and sizing it for my 16GB 9070XT:
Total VRAM: 16GB
Model: ~12GB
128k context size: ~3.9GB
At least I'm pretty sure I landed on 128k... might have been 64k. Regardless, you can see the massive weight (ha) of the meager context size (at least compared to frontier models).
To extend your point: it's not really the storage costs of the size of the cache that's the issue (server-side SSD storage of a few GB isn't expensive), it's the fact that all that data must be moved quickly onto a GPU in a system in which the main constraint is precisely GPU memory bandwidth. That is ultimately the main cost of the cache. If the only cost was keeping a few 10s of GB sitting around on their servers, Anthropic wouldn't need to charge nearly as much as they do for it.