I'd translate it as: "As for now, it's impossible to imagine living without the internet."
You can see 은/는 as 'as for' or 'talking of' - we always say the topic first - it can imply it was possible to imagine living without the internet *in the past*. For this reason, it can be taken as an invisible/subtle 'but, however'. 은/는 always combine with the noun.
In '산다는', 는 is used to make 산다 into an adjective. 는 것 is, as Jae-Woo Kim says, a form that makes the gerund.
As for now, it's impossible to imagine living without the internet.
as for now 이제는
living without the internet 인터넷 없이 산다는 것
it's impossible to imagine 상상할 수 없습니다
What's impossible to imagine? Living without the internet. 을 is used to clarify what the object is.