Can I store the whole blockchain yet keep my actual wallet with bitcoins offline?
Yes you can, 
https://bitcoin.org/en/developer-reference#importaddress

What you need to do is to create a watch-only address on your online computer.
This can be done by "importaddress" in console and use no rescan. Rescan, after you have imported all the addresses to get their respective balances.

To spend BTC from your offline computer, you need to sign the raw-transaction offline( created on your online computer).
Take a look here how-to: https://bitcointalk.org/index.php?topic=651344.msg7306076#msg7306076

If you use Electrum wallet, all these steps can be simplified a lot.