Trekking near Addis Ababa, where can one do it?

Looking on Wikitravel and other trekking sites, it looks like much of the mountains are at least a couple of days away.

However, Entoto Mountain is probably your best bet. From the 'Do' section of Wikitravel:

walk from St. Mary's church, the first church of Addis and St Urael church and see the city itself from the top of the mountain. If planning to hiring a car - Never pay more than 300 birr for a taxi unless you hire them for the whole day and are making other stops. Walking is a great way to make it, but will take a good half day.

So there's a half day walk at least.

From Wikipedia however, it doesn't look particularly taxing. The views in google image search look nice though.

There's also one other mountain mentioned on tour pages - Mt Zuqualla - but I suspect it's basically a guided tour to a lookout, no trekking as such.

It depends mostly what you are looking for. If you seek for some attractions like Entoto which definitely reveals some impressive views over the city you have to know that this kind of spots attract tourists as well as stubborn bagging kids and scammers. A less annoying little highlight might be Wascha Mikael, the blown up stone hewn church of Addis.

If you are looking more for a hiking adventure without a certain destination you may enjoy Addis pretty much. The city is surrounded by many small villages which are connected through little paths. Many of them have been mapped in Open Street Map in recent years. Best way to start is to download the map to a mobile or GPS and go for a trip. Just start slowly in the morning at e.g. French embassy or Gured Shola with a 1-2 hour tour and figure out what is working for you. People are usually friendly and most of the time surprised to see strangers. The security around Addis should not be any problem but I would recommend to be back at least one hour or better two before dawn. A worthy destination for a more extended trip might be the hyena caves.

Last but not least a trip to Menagesha, Ethipia's first national park, is also worth visiting. But a 4x4 is required to reach the park entrance which is located to the opposite of Addis.