If an attacker were to dig up the 'outgoing feed' from my local exchange or the street cabinet, could they sniff my entire neighborhood's network traffic? Or is this infrastructure secured against this sort of attack?
In practice, though, this is a complex and very risky attack - probably going to be easier to do something else.
Yes, they could.
It wouldn't be terribly hard: those street cabinets aren't very physically secure. At least, the ones in my neighborhood aren't. On the other hand, there's some chance of detection (a neighbor might call the police if they see some suspicious character messing around with the cabinet late at night), and there are probably easier ways to make money. Meanwhile, law enforcement and the military have other ways to tap your communications. So this is probably a low risk. There are probably other, better things to worry about.
But, if you are concerned about the risk, the defense is to use traffic encryption. Use an encrypted VPN; use HTTPS Everywhere; use SSH; etc.