The rifle used in that movie was a Lee Speed Metford sporter (I think that I have it all in order), not an Enfield. The Metford was the rifle that Lee developed before the Enfield was designed. You aren't going to find a Metford action to use....the last Metford sporter rifle in fine condition that I saw on Gunbroker went for about $3000. Some of the makers of those rifles were Rigby, Holland & Holland, and the other big name British firms. Rifles in those days typically had longer barrels than the rifles of today, most of the Metfords I've seen on-line had 26"-28" barrels.
You could use an Enfield action to build a respectable copy of the rifle, good enough to give you that "African" feel of hunting with a classic British rifle. The hardest thing nowadays with that is finding an action with a barrel that's not shot out. I've bought 6 different Enfield sporters in the last year, and only two of them had what I'd call a decent barrel. The other four were crap, and wouldn't shoot worth a damn with anything.
You can get a properly sized .303 bore barrel from Montana Rifle Company, in lengths up to 28", but I don't think that they'll fit a barrel to an Enfield action. They're about the only barrel maker who makes the proper bore size for a .303. You can use a regular .30 cal barrel that's .308 bore size and chamber it to .303 British and use common .308 bullets if you're a handloader.
There's a guy out west who has a little company called Special Interest Arms that's an Enfield freak.....he loves the gun. He can do whatever you want done, and the prices aren't bad for what essentially a custom rifle. He's got nice walnut stocks, composite stocks, and laminated stocks. It'll take a while for anything to be done though. I sent an Enfield out to him once, to see if it was worth making into a gun like what you're looking for. He had the gun for three MONTHS, and hadn't even looked at it yet. I told him to just forget it and send it back. That took another 6 weeks and a letter threatening to contact the BBB and his local Chamber of Commerce. So I can't say that I recommend him, but he can do the work you want done.
You definitely want to find a gunsmith that's comfortable with Enfields to do the work, there are a lot of little details with the Enfield action that aren't like the common bolt actions of America. There are different size bolt heads for head-space issues, the barrel has a shoulder milled on it that can make removal difficult without a lathe, etc. It's also extremely easy to bend the action trying to remove the barrel.
I've decided that when I have the money for the project (in other words, money that my wife is un-aware I have

), I've going to have a Ruger #1 single shot done up in .303 British to resemble a single shot sporter similar to that time period.
If you liked the movie, the actual book of the true story is called "The Man-Eaters of Tsavo" by Lt. Colonel J.H. Patterson, and it's a great read. I got the book for CHRISTmas several tears ago from a buddy who knew that I loved the movie.