By stake it means Algos, not any other kind of hardware resource since doesn’t rely on Proof of Work consensus but Proof of Stake. I’d recommend taking a look at this video from Silvio.
So yes, if you don’t have Algos, you’ll not get rewards.
If you purchase Algos or get them in future auctions, you can transfer them to an account you own. To understand the concept of online/offline, I think you can check this thread where @JasonW replied to some questions.