Why has my homemade limoncello got a cloudy residue at the bottom?

It is just lemon particles falling out of suspension. These things take time and the particles are too small to easily filter out, so next time just leave the whole thing in your big container until the Limoncello has cleared up a bit, that way your bottles will look nicer.

It may take a very long time to completely get all the particles out of suspension, so you may want to consider how much effort you want to put into this. You should of cause give it a try and wait as long as your comfortable with, but I've had home made wines that looked clear, and even after half a year of waiting to bottle there was still some particles coming out of suspension and giving my bottles a super thin layer of sediment

I have made 4 or 5 Limoncello (Lemon and also Lime based) batches and every one of them has been hazy. The last one has (or had until I skimmed it) a layer of oily flotsam - I did taste it - bitter an oily - I am guessing I got waxed lemons by mistake - but it didn't ruin the the actual liquor. The haziness always sets in when I add the lemon juice.

I tend to add the sugar or honey to the spirit and then add the finely pared skins (not taking the pith). Then juice the lemons and freeze. Turn the jar every time you are near it, don't shake. At around 5 days, when the sugar is dissolved and the oils extracted, defrost and add the juice, strain the lot into bottles.

Shop bought Limoncello (at least the ones I have access to) are generally hazy too. I wouldn't worry unless it tastes "off". If it is full-on roiling clouds - it is probably bits from squeezing the lemon. Let it sink and decant.