The Iron Spike Model Train Museum is in Washington, Missouri at 1498 High Street.
You can actually take an Amtrak train from St. Louis. There's a shuttle that will meet you at the station and take you to the museum and other area attractions. The museum is open Friday, Saturday and Sunday or during the week by appointment which you can make by calling 636-283-5166.
You can find more information about the place at ironspike.org.