Seeing this is a 'suck it and see' job, I suggest you start by reading Tom Deakin's article
Fixed Gear on the Cheap - it's part of Sheldon Brown's series of articles on fixed gear and that resoure is THE place to start. Read it all.
My Europa started as a simple suicide hub conversion - ie, I took off the gear bits and pieces, spun on a track cog, redished the rear wheel (actually, I didn't have to redish the Europa's rear but I've done it on other bikes, it's not a big deal) and went fixed gear riding.
The thing about a suicide hub is that you have to do up the cog really, really tight (a chain whip isn't enough) because the cog can come undone under back pressure and that's ... exciting. I used to install the chain, put a block of wood betwen the crank and frame, then use the wheel rim to pull the cog on as tight as possible. While some condemn it, it's really only a problem if you use your legs for braking and try skids and the like. I used the brakes and moderate back pressure so it was never a problem for me.
That's the cheapest way of trying it out.
I converted the Europa, found I loved it and THEN went looking for wheels. Rather than
ebay, I was able to get a local wheel builder to build me some cheap wheels - novatec hubs with some unknown chinese brand rims - those wheels are still serving the old girl well. The Europa's got to her current state over a period of years, slowly losing old bits (like the spare chainring) and gaining fixed specific bits.
700c wheels will go into your frame easily enough, but the rims are a smaller diameter and you may find your brakes don't reach. Generally, old roadie brakes will reach and Tektro make some very long reach, dual pivot brakes at a decent price so brakes needn't be a problem. This is where you need to cultivate your local fixie shop rather than assuming that
ebay will always be cheaper (it's not and it's not as convenient).
You can still get 27" tyres and tubes but they can be hard to find among shops that pander to the modern industry rather than looking after cyclists in general. However, converting to 700c opens up the whole market of tyres.
My recomendation is to spend a happy saturday arvo doing a cheap coversion as described in Tom Deakin's article. This gets you on the road to try it. Once you discover you love it, gradually convert her, using decent parts as you go.
Richard
I had a good bike ... so I fixed it