By “internet” do you mean anything using the various internet protocols, which really is a problem since the internet is simply the system using the telecommunication system. By that definition, “internet” includes telephone, the entire banking system, all credit cards, and virtually all sources for hearing news. Not using the internet means living with an 18th century level of technology.
If by internet you are referring to “graphic browsers” one could exist without them, using PINE mail, LYNX and not much more. You would probably lack access to the banking system and telephonic communications.
If you say the objection is to certain websites, there is no hiddush. As older people may remember, it is prohibited to look at undressed people acting improperly even if it is offline. Obscene literature was probably invented about the same time as writing – it isn’t a function of the internet.