I, or rather Bicycle Doctor in Manchester, did this to my bike - it did require a new rear wheel and cassette, STIs, new rear mech, the rear triangle of the frame cold setting some adaptors brazing on to the down tube as cable stops (as my friction shifters were band on). If you want to go from double to triple then you'll need a new chainset too.
It is a big job and unless the frame is worth keeping (mine was a fairly nice Bob Jackson) and some/all of the things you're replacing are close to the end of their useful life anyway then a new bike will be the way to go.
However, one thing I would say is I made two very expensive mistakes on mine; I went with Sora shifters to save £50 (or something like that), I hated the little thumb toggle for downshifting, I eventually found a pair of 105s on
ebay secondhand for about £100 plus paying an LBS to fit them. I also didn't bother getting it resprayed (just hammerited the brazed bits) again to save £50-£100, which was pointless as when I eventually got it resprayed the biggest cost was getting it stripped and rebuilt - which pretty much had to be done for the STI upgrade anyway...
So if you do have a very nice frame then I would advise going for at least 105 for the shifters, getting it resprayed at the same time and possibly learning to rebuild it yourself (something I've now done). I don't expect you'll get away with it costing much less than £500-£700 which is quite a lot given that a new bike with a full 105 groupset can be had for around the £1000 mark, but if its a nice frame then it might be worth it...