Sorry, I can't get the image of those little ride-on steam trains you get in kids' parks out of my mind. Maybe some eccentric elderly men who liked having an uber train set to play with (under the guise of providing a 'service' to the local recreation department) left behind their model railway and the Hobbits later found it - of course it would be 'just their size'. Meh.
Hmm, thinking about Saruman and how far he and others had come in terms of technology, a steam train would not be entirely out of the question (though I could not place one in The Shire at all). Gandalf had developed some advanced ballistics with his Fireworks, Sauron seemed to be using advanced weaponry systems st the Battle of the Pelennor Fields, including flamethrowers and petrol-based ballistics. Saruman was already on to using gunpowder in weaponry, and Gandalf must have known about its uses too - maybe Boro, our fireworks expert can clarify whether fireworks must use gunpowder to work?
Historically, some developments were rapidly followed by others. Hmmm, my memory of the exact inventors is rusty, but I think it was Newcommen who developed the steam engine and it was a mere handful of years alter that the forst railways were being built. What, exactly, did Saruman have in operation down at Isengard?
There is also the possibility it refers to other kinds of train. That East Road sounds suspiciously well surfaced to me what with all the quick movements along it and the existence of little 'toll houses' when the Hobbits return at the end of RotK. Turnpikes? Maybe!
And of course there is the fact that The Shire seems suspiciously much more advanced than other societies in Middle-earth, what with its umbrellas, clocks and egalitarian system of Government. It's at the very least reflective of a regency village society and has a lot of the Edwardian too. It's certainly not medieval!