Back in 2009, anynone with a Nokia could have a personal website running on their own phone. Sadly this amazing piece of tech was never widely adopted. Today’s phone are far more powerful than those Nokias both in performance and battery backup and still we don’t see anyone running a server on their phone. Why?
I think this was never implemented on phones because there’s no incentive for large corporations to work on something like this.
Assuming you’re hosting on a cellular network, but actually to run a small site on a local network connected by wifi it would be an option. Not necessarily the best option, but doable and yeah the battery life could be useful for intermittent power situations.