unless your website has very limited storage allowances or you have gigabytes worth of videos, you should be able to store your videos on you website server.
As for Web site file storage and bandwidth allotments, it's gotten a lot cheaper than it used to be. Check out prices for basic hosting plans here for just one example of low priced hosting:
Great company that I have used for many years and have many hundreds of videos on various sites. As you compare, look for features like unlimited storage and unlimited bandwidth... it's not that expensive.
You mention mpeg video files... mpeg1 or 2... not mp4s? so the files are huge? and the files will not be viewed on the Web site but rather downloaded to viewers machine and then played back?
If mpeg1 or 2, the viewer will have to download first so you might want to use a little PHP script to allow the viewer to select from a list of video files and direct download those they want. Here is an example of video files for download... lots of video files, to give you some idea of the amount of videos that can be stored on a site... GBs and GBs worth!
the page uses a simple little link like this:
<a href="downloads/process.php?file=judy.MPG">Download judy.MPG</a>
to a small "process.php" file. Click a link and test the download for yourself.
Here is the php file used:
$file = $_GET['file'];
header ("Content-type: octet/stream");
header ("Content-disposition: attachment; filename=".$file.";");
Save this file as "process.php" and place in the same folder as all the video files and link to them accordingly.
However, if the files are mp4 and you want to allow visitors to view the videos on line, then you need a media player that can handle those particular type of files. Build one yourself or use something like the JW Player:
Then use your Flash to display your other content.
Best of luck on yur project!