Yes, you certainly can. There are numerous ways you can go about implementing this. I shall be listing 3 major ones.
Using Bitcoin/Ethereum Protocol:
Since both ethereum and Bitcoin are open source, you should be able to obtain their source code, tweak it in a way that it doesn’t attempt to sync with the Main bitcoin/ethereum protocol but rather your very own private net where you’d essentially be running a similar version without being synced to the rest of the nodes throughout the world. You could either run it in virtual machines or in containers across different servers, but it isn’t quite efficient or cost effective.
This however requires a fair amount of knowledge and is quite complex as well, thus isn’t recommended unless you’ve got the required expertise and time.
IBM hyper ledger is another option where a commercial version would be quite expensive, but you can run server nodes to mine your blockchain directly without involving a lot of people. You may single handedly build, own, mine and run the blockchain. Hyperledger is suitable for consortium and private blockchains where your information has to be private.
This is fairly easier than modifying the Bitcoin / Ethereum Protocol, but quite expensive for commercial purposes.
Multichain is an open source platform to build blockchain applications directly.
Multichain is fairly newer comparatively but is developing traction at an alarming rate. Multichain also comes with the ability to create smart contracts in the form of streams. Multichain is both open source and free making it a viable choice for hobbyists to get started with, however the lack of support from a dedicated team could be a downside. Multichain also allows you to run multiple nodes on the same machine.