Is it feasible to use the Blockchain technology without the Bitcoin? Every once and a while, I step into some article like this one: Forget Bitcoin -- What Is the Blockchain and Why Should You Care?.

I've been hearing a lot of blabbering about companies to begin using the Blockchain technology for book keeping, for example, but making no use of Bitcoin at all.

I wonder: how is it possible to separate both things?

I mean, the aim of Satoshi Nakamoto was to create a decentralized coin that would supress the need of a trustworthy third part, which operates over the Blockchain.

The main aspect of the Blockchain is that it's a decentralized network, working on a peer-to-peer basis, no servers, no clients, only peers. The only reason someone would contribute to the building of the Blockchain is the monetary reward in form of mined bitcoins.

How could a company make use of the Blockchain? How would it pay someone to do its book keeping (or whatever) without payments in crypto-currency?

I see that this might be a rather wide question, but I'd be satisfied with only one example of how could it be possible to use the Blockchain technology without Bitcoin (or similar crypto-currencies).
Blockchain by definition is a data structure, in that sense we can and already have decoupled it from Bitcoin.  There are many alt-currencies that implement blockchain as a data structure and a data store.  

Ethereum is building a blockchain based application platform.   

Storj is an example of blockchain technology that also is trying to create a blockchain for data storage. 

The SAFE(Secure Access For Everyone) network is attempting to build an entire computing platform built on a blockchain.  

The key is incentivizing some form of participation by providing a service. On the Bitcoin blockchain the service being provided is the ability to send value globally very quickly.  Thus the reward that was created is financial.