In current, real-world networks, what benefits would physical access to a networked machine provide a potential attacker?
The most obvious example is an 'air gapped' network that is not connected to the internet - there was a real-life example of this in stuxnet where the target facility had no external networking. To physically access the network, the attackers dropped USB drives containing the worm around the facility that facility employees found, picked up, and plugged in, accidentally releasing the worm into their internal network.
You may also frequently hear the term "m&m network" or "m&m security". These terms refer to a network with a hard 'outer shell' and a soft 'chocolate center' - networks that have strong perimeter security but no security features (encryption, authentication, etc) between services or hosts in their network. Physical access to one of these networks can help bypass the hard outer shell.
How do these benefits compare to the risks associated with breaching physical security at, say, an average data center?
Like other risk questions, this depends on risk/reward. If all you need to do is drop some USB drives near the target network there is low risk. I've been in a few data centers and they all have substantial security - unauthorized physical access is highly unlikely, and employees have been trained against common social engineering tricks and won't disclose information or plug in USB devices. Suffice to say that 99.999% of the time you can be assured that nobody will have unauthorized physical access to the datacenter.
Finally, are there any network security measures in use today which can effectively prevent remote access by an attacker?
This is too broadly scoped to fully answer, but a good example is the airgapping we discussed before: disconnecting your network from the public internet to prevent anyone remote from even seeing your network. Of course this has the drawback of preventing your network from talking to any other networks or the internet.