How to reward mining in a blockchain that doesn't store currency? I understand the bitcoin blockchain motivates mining because there's a bitcoin reward for the miner who solves the pending block.

What if a blockchain doesn't store currency data but some sort of ownership data that cannot be represented by currency, e.g. tweets or land and stock ownership.

It would probably be too much to reward a miner with the ownership of land. And even if it were appropriate, there could also be other cases in which the blockchain doesn't store data regarding any kind of asset ownership.

Is it possible to offer reward that is not coupled to the data of the system?
There are several blockchains that already do this. Colored Coins (on bitcoin blockchain), Counterpary (on bitcoin blockchain), NXT, BitShares, Ethereum and others allow to represent any asset you wish like land, cars, consultancy hours, etc., using digital tokens (e.g. the "data" of the system as you called it) which is irrespective/decoupled of the reward they get.

See Colored Coins for how you can use metadata in a bitcoin transaction to achieve that result. It is especially easy to implement using Counterparty.

So, you can already store the kind of data you want in parallel to a digital token that rewards the miners.

If you meant, only having the asset in the blockchain and no reward then the answer is no, you cannot: you always need something in digital form that can be securely shared to the contributors**.

Hope this helped.

**For an internal reward, you have to give something of value to the contributors within the system. Say, that you don't want to give money/currency but instead some service. In order to keep a proper count of the rewards for each contributor you would need a digital way of representing the reward, ie. by using a digital token. To securely keep track of the rewards you need to also put them in an immutable structure, ie. the blockchain.

Thus, whether you want to give money or chickens or some kind of service to the contributors you have to digitally represent it (tokens) and secure it (blockchain). The token represents the reward and it does not need to be money. However, it is very practical to think about it as money since this is the unit of exchange that most of us are used to.