I have two new-ish 1TB hard drives in my server, both running at 38 degrees C at the moment. Should I be worried?
38C should be fine. A quick look at some spec sheets shows that operating temperature is 0C to 60C on many models.
You might be interested in section 3.4 of a study done by Google on hard drive failure rates. Temperature is not an issue in drives until they are a few years older, and even then, it seems, it's not much of an issue under 40C.
Be careful of the interpretation of your sensors. I have Munin running, and the smartd results shows my drives have "Temperature_Celcius" of around 230.0 (which if true would probably indicate they were on fire) but another probe called "HDD Temperature" records them around 27 and 29 C, which seems much more likely to me.
HDD can run in a range of 0-60°C, but it is not optimal. They can function with reduced lifespan at either end of these extremes. prolonged exposure under 20°C or above 50°C will cause impairment of the drives useful life. The safest is between 20-45°C. So 38°C is somewhere in this range. 30-32.5°C would be ideal.
One test by national instruments found that a 5C increase could reduce the life by 2 years.
More info here :- https://www.akcp.com/blog/how-temperature-affects-it-data-storage/