Apart from recording somewhere quieter...

I tend to use Reaper rather than Audacity if I record things just because I find it better. One of the many things on that when recording is to put a noise gate on the track when you are recording which stops some of the low level hum and background rubbish appearing so that you don't have to try and remove it afterwards which, as said above, tends to affect the whole track.

It does at least mean that in the quietest bits or in in 'silence' that you can cut the background sound out - which is where you tend to most notice it.

Audacity have one too - Noise Gate - so it might be worth doing some experimentation with it. You need to get the right balance so that things sound smooth but it may help. If it doesn't get there in the first place you don't have to remove it!