Most people already have an MP4 player. It's called a mobile phone (which usually has a far better screen than a dedicated media player anyway).
However if you want to find a standalone device I suggest that you specifically search for an 'mp4 player' (rather than for a 'media player', or whatever). That comes up with things like this:
http://www.argos.co.uk/product/9082177
Many MP3 players don't even have a screen so, clearly, they're not going to play MP4 video files. (My MP3 player, that I use in my car because it has a built-in transmitter to send a signal to the car's radio, looks like this for example: https://tinyurl.com/yac2ew3a )
However I'd expect most MP4 players to be capable of handling MP3 files (but I'd advise checking on the specifications before making a purchase).
I have one of those Phillips player that Buenchico supplied the link to, it's a marvellous piece of kit and although I have already loaded loads of music onto it, there still seems ample of space left.