Exploring a local copy of the Bitcoin blockchain without running a full node? I've been hitting the rate limit for API calls for different services and have decided that it would be easier to have a local copy of the blockchain and query that instead. See my previous question:

Get fees for each transaction in a block without too many API calls

Is there a way to keep a continually updated copy of the blockchain locally?

Any suggestions for block explorers that would work in this manner?
This sounds like you'd like to run a full node, which does exactly what you are requesting. You can run a Bitcoin Core full node with wallet services disabled if you want to use it just for blockchain analysis. I'd suggest following along the bitcoin.org guide to running a full node, and take a look at the Bitcoin Core Config Generator for all the different configuration options. You'll need at least 180GB of free HD space, especially if you are going to run the node with the Transaction Index turned on, which you will want for doing fee analysis.