The scenario you describe could be the result of highly coordinated attack by an ultra-radical anti-technology group that has managed to simultaneously attack and knock out every Tier 1 network provider and every Tier 2 provider on the planet. I'm exceptionally impressed with their ability to pull this off. They have destroyed the world.
One could stage attacks on the BGP protocol used to control routing between large networks but attacks of this kind wouldn't result in a long term outage because the network operators are very careful to make sure their networks stay up....after all, that's what their customers are paying for.
Tl;dr (2022 Edit)
Remember when COVID hit and most things just stopped or slowed way down? The situation in OP will feel similar for a day or two then keep getting much much worse. How you felt then will repeat here but much much worse and for a lot longer.
Every router and switch at the Tier 1 and Tier 2 network providers locations are gone. Millions of devices have just disappeared. Vanished. The wires are just dangling there.
People will respond with "Hey, is the internet working for you?" "Nope, I can't get to Google or Facebook." Internet addicts will start to feel the first pangs of withdrawl. All phone calls halt. Financial institutions who depend on Internet connectivity to clear transactions will start to feel the pain. High frequency trading companies will absolutely lose their minds, as will every engineer in the T1 & T2 network operations centers. In an instant, untold billions of dollars of investment have vanished. While the ability to build new routers and switches remains intact, the money to buy them is gone. New switches and routers are suddenly priceless and worthless at the same time. Priceless because every network operator on the planet wants them. Worthless because without another router on the other end, a single router can't do much.
The electrical grid which has been smartened up no longer functions as well because coordination between power providers cannot occur. Having the load from the switches and routers disappear may cause damage to steam generators as they will suddenly overspeed. But other power generations sources will enjoy the decrease in load.
Remote surgeries will initiate their Internet-failure protocols to close up the patient safely. Surgeries where the surgeon is on site will continue, though who knows when/if he'll get paid for his services.
Internet addicts are having fits as is every single teenage boy addicted to the World of Warcraft (or whatever online game Blizzard is running). All commerce has stopped. Credit cards don't work. Let's assume that cash has been replaced with cryptocurrency....which also no longer works because to clear a cryptocurrency transaction requires an internet connection. Stores will be able to sell things using the very primitive "CHUNK-CHUNK" paper based records. They won't be able to clear these. Every single stock and commodities exchange on the planet has frozen. Packages in transit over UPS will arrive if they are already on the truck but no new packages can be shipped.
Ham radio operators begin to share information about the outage. They describe the scope of the outage in their area and over the next couple of hours the scope of the outage begins to take shape. Mesh networks (which hopefully are widespread at this point), kick into high gear to share information about what's going on. Connectivity within a town or city may be high but connectivity between cities is zero. Mesh networks can't handle that kind of load.
Millions are stranded at airports because boarding passes can no longer be checked. It's like 9/11 again, only it's everywhere and no one can pay for alternate transit. Container ships with the goods of civilization arrive at port, are unloaded but the containers just sit on the docks because no one can figure out where to send them with no way to call to ask someone.
Mayors, governors and heads of state declare a state of emergency but not many people can hear them.
Panic has set in. Any kind of a store with food is now empty. Violence escalates as people begin to forcibly take the things they think they need to survive.
Factories that build routers and switches will go into monster overdrive as demand for new routers has gone through the roof. It will take years to manufacture new routers to satisfy the demand. Businesses of all kinds have shutdown or are in panic mode, hospitals too.
Riots are in full swing. Police and fire departments are stretched beyond breaking. Vigilante justice and neighborhood protection groups spring into existence.
The riots and chaos of the last week have largely subsided. The barter system has returned in force. Small items are now used as currency in place of money or a local cryptocurrency has sprung up, supported by mesh networking. Communities have shrunk and become geographically focused. People have met their neighbors for the first time in their lives. Anyone with a mesh network node is extremely popular. Anyone with mesh networking expertise is extremely valuable. Google and other Internet companies have lost significant portions of their stock value. Phone manufacturers have also lost practically all value and will close or be supported as "too important to fail" and kept on nationalize life support till the market comes back.
Medical and food supplies are running low because there's no way to reorder them.
For a better idea about what happens when commerce significantly slows in a country, look at the changes in Greece between April 2015 and mid-July 2015. This situation is worse because commerce has stopped, where Greece has only slowed down (a lot).
Demand for Flash drives and external storage media go through the roof as people fall back to moving data around by hand. Postal and package service may resume service but with significant delays.
Every single business that staked their productivity on Applications-In-The-Cloud are having a really really tough time. They can't do business and won't be able to do business for a long time. Secondly, they may never be able to get their data back because the companies that hold their data may go out of business and take the data with them.
Internet providers are nationalized or placed under centralized government control in order to manage the trickle of routers and switches coming out of the factories. Companies have fallen back to the old paper forms they used to use the 1980s. Snarky gits laugh at the organizations who went all paperless.
P2P application such as BitCoin and BitTorrent become the primary means of or paying for things or distributing information. These applications work because they don't need any centralized source to run. On a network of three computers (with the appropriate software) these are completely functional networks.
Open source software (distributed source control, compilers, editors, operating systems etc), already powerful in their 40+ years of operation, step up to fill the gaps in services left open by the disappearance of the large providers such as Google, Facebook, Apple and Amazon (and whoever else may pop up in the next 20 years.)
Internet access is slowly returning but the quality of service is significantly slower than it was. Shoddy equipment causes datacenter fires, setting back the return of internet service for months or years.
Ham radio becomes an unprecedentedly popular past-time, causing a boost in science education.
The mints of the world have been de-mothballed and begun to turn out hard currency again. Designs of old currency are renewed.
Culture in general will go through a period of introspection on whether Instagram, Snapchat, Facebook, are really all that important anyway? Many people may say "No, it isn't" and find other things to do.
Merging all the cryptocurrencies that have sprung up back into a single national currency becomes a legal nightmare.
The courts are absolutely chalk full of lawsuits. Legislation will need to be written to cut down the number of lawsuits and handle the aftermath of the Outage.
Internet service is back for everyone. Network topologies rely increasingly on mesh principles. Large telecom providers have been bailed out. Market share in the network equipment space is unrecognizable compared to pre-Outage as new companies appeared to fill the staggering demand for new routers and switches. Cryptocurrencies become more tolerant against widespread disconnects.
Mints and postal services the world over issue 10 year commemorative coins and stamps.
Some people will ask "Where were you when the Outage hit?"
While the dedicated routers may be gone, the ability to route Internet traffic hasn't completely disappeared, it's just grown horribly more inefficient and lower capacity. Routers are just computers with lots of network ports and a specialized form factor to better fit into data centers. The only difference between a router and a general computer is the number of network ports and the software running on the computer.
There are also long distance microwave transmitters that can move traffic tens of miles at a time. Long distance data trading could happen this way.
Inventive network operators will begin to repurposing multi-port servers to be routers. The capacity of these machines is far below what used to be available but at least some data will flow. Prioritization of traffic in these early days will be intensely political.