I have a 2004 Matrix with 80000 miles on it. In the last couple weeks, I’ve been getting some odd engine noise. It’s particularly noticeable when I accelerate from a dead stop, either in forward or reverse. The noise sounds at first a bit like a diesel engine (though not nearly as loud) and then smooths out into a sound like an oscillating fan that won’t quite oscillate, if you know what I mean. Again, these aren’t very loud noises, but noticeable when the radio isn’t on. 
Any ideas?
Your description sounds a lot like detonation or pinging. Are you familiar with how those conditions sound?
Both conditions are undesirable and have a number of potential causes. Long term, it can damage your engine and if it’s severe enough, it will be damaged much sooner. If you’re reluctant to seek professional service, one test you could try is to use premium gas for a tankful. If that reduces or eliminates the noise, this is your problem. Either way, it should be addressed.
What kind of repair is necessary to fix this problem?
what kind of fuel are you using?
usually you get this from using fuel that is too “low-grade” for your car. find out what the manufacturer recommends and start using that.
Here are some examples of potential causes; incorrect ignition timing, carbon buildup in the cylinders, need for “tuneup” like plugs and wires, lazy O2 sensors etc. The need to bump up a grade or two on fuel octane is usually masking one of these type of problems- assuming you’re using the recommended fuel for the vehicle.