All the humans disappear, except maybe less than ten (unskilled) people.
How long would those few be able to access and use the internet?
Not very long at all.
Probably days at best. While I'm not sure about the infrastructure of the internet itself (which I think would probably last a little bit longer before catastrophic failure) the more pressing issue would be power. Without people running the power plants and managing the grids around the world the electricity would soon go out.
No electricity means no computers, so even if the few remaining humans are lucky enough to have a generator or be somewhere the power has lasted longer, most of the internet will have gone offline when the servers died due to lack of power.
The answer to this is very dependent on geography, both of where those 'unskilled people' are and what part of the Internet they're trying to connect to.
Power: As many people have noted, as soon as the Great Disappearance happens, electricity will start to get spotty. Even if all vehicles avoid causing outages via downed lines, unchecked fires, freezes, storms and etc will start chipping away at the power grid. That said, cities are slightly less vulnerable than rural areas because of better protected, higher-density infrastructure: power is often underground instead of on a pole, and a denser grid means more redundant routes to the power station.
Networking Hardware: The core routers of the internet by now mostly reside in datacenters designed for 24/7 redundant operation. If their power fails - see above about those possibilities, and figure that a datacenter in a major city could indeed have feeds from multiple power substations - they generally keep a day or two's worth of diesel around, as that's considered long enough to get more trucked in in the event of a major outage.
Networking Software: The Internet is designed for redundancy. As long as you can reach a defaultless router (one that contains the core routing tables), you can reach anyone else that can do so similarly. And you'll have plenty of bandwidth to do so, as those that disappeared will no longer be using all the interconnect bandwidth by streaming netflix and youtube (which together are over 50% of downstream network usage). Failure after that will start by breaking the network off into isolated sections (which can still be used to talk among themselves!), until eventually it's all in so many pieces that it effectively no longer exists.
Overall, my estimate would be days (given decent weather) in rural areas and weeks in urban areas.
Within hours, Lights around the world will begin to shut off. any servers which don't have backup power will go down. Even then, the remaining people won't be able to use the net if they don't have backup or can't help in producing electricity.
Those which have backups might work for a year, after which most satellites will fall back to the Earth. So you can say the max is 1 year, if both the servers, medium and the people survive.
If you consider only the backbone, the actual transmission of data, the internet basically becomes IP addresses. As long as there are two computers, and routers between these computers, that is an internet. Think of your home router. How much attention does it need? Sometimes, it needs a reboot. Otherwise, it will keep chugging along until the power goes out. Same with the internet backbone.
Perhaps the biggest job of IT technicians is to reboot recalcitrant routers, and dealing with hackers. Replacing the occasional switch that fries. Adding capacity. Otherwise, they just spend most of their time on customer support, telling THEM to unplug and reboot.
The life of the backbone then becomes an issue of back-up power. Because of the expense, I can not imagine anything with more than a few days of back-up power, more likely hours. Even back-up generators have a finite fuel supply, and need someone to continuously refill them.
But if you have a local system of routers, with batteries and solar back-up, the system should remain live for years, until either the router needs a reboot, or the electronics gives out. Most electronics is good for at least five, more like ten, twenty, even thirty years. Eventually, it is usually heat that destroys it.
Satellites, as we have seen in the Voyager system, can keep on ticking for decades. Their weak point is in keeping the sending Earth-based antennae directional and lubricated. Yet, people have home dishes pointed accurately at geostationary satellites for over a decade. The biggest weak link for these satellites is in the transmission points sending the data in the first place. Consumer dishes are receive only. They will continue to receive, as long as there is something left that is transmitting.