Will my system be secure?
Maybe. While you say you do not browse from the system nor does it communicate with the internet, there are a number of other possible scenarios to contend with:
- That the system is already compromised, making this discussion moot. Unlikely, but still possible.
- That the system can be attacked from compromised hosts on your internal network.
- The type of attacker you are defending against; e.g. a determined targeted attack ("advanced persistent threat") or a scattergun approach.
Whether the second point holds really depends on your configuration and whether any holes exist in the OS once it reaches end of life. If there are, these will not be fixed.
Whether these holes can be exploited really depends on the type of attack. An automated scattergun-type malware might optionally attempt to scan for known vulnerable systems and try to compromise those too. Or it may miss them, on the basis of no longer considering them viable due to dwindling install numbers.
A targeted attack is likely to be different. If I was looking at how to attack your system, and I had knowledge of this box running beyond-the-grave-software, I'd be cackling like Borris from goldeneye (pleased, basically). The reason for this is simple: if I can find a hole on this box, I know it won't be patched. I no longer have to deliver an exploit within a certain Window - I can simply compromise at will. Got any other systems on the same platform? Great, I can attack those too. Even if you re-install, I can re-infect - because plugging the hole will not happen.
As such, I'd tend to err on the side of saying no, it's not secure. There's a lot of 'if's in that assumption, and a lot of adjustment depending on the risks you believe you face, but given the machine is a fairly critical one, I'd update.