Since recording movies of real trains is pretty much a non-starter (too many of them already, pics and otherwise), I think he's talking about small clips of model railroad trains moving around a built "landscape".
Wasn't the question more "How can I make a site to upload my train clips to?" Not which host can I use.
If you are newbie with web page design, then try to get a host with Fantastico, or some other autoinstaller program, because there are scripts where you can build 'communities'- with upload movies functions etc. That means you don't have to do anything technical.
youtube.com or a similiar video site (many of them out there) if its just small clips (i've seen clips as large as 30 minutes on youtube).
If you insist, or have a real reason, to have your own site, you can get a VPS or dedicated server, hire PMS or some other management company to handle the server, and use a youtube-like script (i've seen some around) to start your site.
You might can even get by without management if you have WHM/cPanel installed on the server. This makes things very easy to manage for most things.
If you have a little money to spare, try posting in one of the request forums here on WHT. I bet you could find someone to install a youtube-like script and handle basic server setup for a pretty reasonable price.
Some people may say shared hosting. This might work, but if you cannot use youtube for your needs - chances are your resource usage would be too high for a shared hosting plan and you would not last long.
Last thing i'll say is do not over compensate! Don't order 4000 GB of bandwidth (or unlimited) just because you are hosting a video site. Start with a reasonable number and upgrade as needed. For a VPS, the reasonable number might be 100 GB bandwidth (although most providers oversell and offer way more than that).
For a dedicated server, you'll likely end up with 1000 GB to 1500 GB of bandwidth a month to start with.