The biggest thing that brings down a site is changes. Typically code changes, but also schema/data changes, infra/network/config changes, etc. As long as nothing changes, and you don't run out of disk space (from logs for example), things stay working pretty much just fine. The trick is to design it to be as immutable and simple as possible.
There are other things that can bring a site down, like security issues, or bugs triggered by unusual states, too much traffic, etc. But generally speaking those things are rare and don't bring down an entire site.
The last thing off the top of my head that will absolutely bring a site down over time, is expired certs. If, for any reason at all, a cert fails to be regenerated (say, your etcd certs, or some weird one-off tool underpinning everything that somebody has to remember to regen every 360 days), they will expire, and it will be a very fun day at the office. Over a long enough period of time, your web server's TLS version will be obsoleted in new browser versions, and nobody will be able to load it.