Let's start by addressing the "leave as much other technology intact as possible" part first:
10 years ago, you could have taken down the internet and most things would have continued to function just fine, but today, almost everything that people use most has some sort of cloud based element, that if removed, will cause your software to break. The Office 365 and Adobe CC products which most people use for the majority of their multimedia reading and writing applications can only go in offline mode so long before they will fail to authenticate their license and stop working. Most top tier video games these days are some manner or MMO, and those that are not still often have similar Digital Restrictions Management components to the aforementioned microsoft and adobe products. Many ad-funded freeware applications that can not find their ad server will crash as they wait indefinitely for content that they can not reach.
But this inconvenience in your personal life is the warm and cuddly aspect of what no internet means.
The truly scary part is that most business grade software relies on decentralized data. Nearly every businesses in the world would come to a grinding halt when they find themselves suddenly unable to access their inventory data, their financials, their client lists, etc. Shipping companies could no longer fulfill their orders, manufacturing companies could no longer receive materials or distribute their final products, most experts would be too young to know how to fall back on traditional skill sets to get the same jobs done, many tech companies could no longer deliver their services, most phones would stop working because all those cell towers have their own DNS addresses, and most land lines are now VoIP. the list goes on, but long story short is that the economic damage would be so extensive that the world would not just revert to a pre-info age state, but it would in many ways revert to more of an early industrial age state because we no longer have the pencil-and-paper infrastructure that first spanned the gap allowing us to get from basic localized industries to globalization... Not to mention with the economy that fully collapsed, there would be massive famines, wars, rioting... things would get really ugly really fast.
Now let's address the outage itself:
It is REALLY hard to knock out the whole internet, and even harder to keep it down.
My original hypothesis of a Root Name DNS Server attack seems beyond the realm of plausibility based on feedback. While there are only 13 Root DNS "servers", their distributed nature and massive redundancy of possible backups makes an attack against them highly improbable to not be easily recovered from.
Another common hypothesis you will see in other answers are solar flares, but these only cause very temporary interruptions. 111
Any flare weak enough to not completely scour the world of technology and perhaps many forms of life will not cause permanent hardware damage to more than a tiny % of internet connections. Once the flares are done, the internet would go back to normal. No big deal.
Viruses also don't work because there are too many different tech stacks at play for any one virus to be able to come close to wiping it all out.
If you are REALLY rich, purchasing all of the world's ISPs and simply firing everyone so that they all shut down would work in theory, but human nature would not tolerate such an unethical choice and the world's governments would likely just seize your assets and charge you with cyber terrorism, breach of contract, monopolization, child pornography, and whatever other real or made up things they can accuse you of to justify seizing your assets and turning the internet back on.
This basically just leaves government censorship. No populus would simply allow their government to completely shut down all cyber communications in one fell swoop, but governments all the time get away with restricting the services their citizens can access under the guise of National Security. Normally this is done by blacklist meaning your government regulated ISPs wont permit you to visit X,Y,Z sites. A more restrictive approach would be to make the internet whitelist censored meaning you can ONLY access sites the government approves. Such a standard could be made even more restrictive if you had to be licensed to use the internet. This way, having internet access at all becomes very rare outside of legitimate business and government applications. If this were the case, the government could force all of the businesses in their nation to migrate to only using approved services, then shut down everything that is not approved. That way businesses could still bank, and use a select few business management tools, but there would no longer be the option for citizens to just add websites to the internet; so, you could effectively eliminate all recreational and social applications of and make the everyday person revert to a pre-internet lifestyle. 207