Imagine if it faulted the same way and no smell. Harder to locate, so in a round about way, you're lucky you have this indicator.
You didn't specify model ML380 (which one). I suggest you check with the HP website for model specifics, and while there, download the pdf's for it. These will give you step by step and pictorial information, especially in the do-it-yourself category.
There's also the question of Warranty for that specific model. If you're the first owner, it's possible you have warranty, or if it's second hand, the sales receipt of the previous owner could be valuable. If no to either, the HP information downloads are still needed to at least confirm parts identification and handling.
Open the case for ventilation.
Pull the power supply and smell it.
Pull the Ram and check it for burned areas and smell. If all seems well, leave the ram aside and check for burned areas of the motherboards surface part area.
Pay particular attention to the standup electrolytic capacitors, that they are not exploded, or their tops are not crescent rounded, an indication of internal gases bulging the part. If this seems OK, (next)
Considering the most likely's:
...it's power supply, motherboard parts, Ram, in that order or the rare combination of either.
If none of these items smell harsh, keep checking.
My bet is on the supply.
I'll be monitoring for your progress. Good luck. Robert
