It is possible to make your own, but to make banners you need a program like GIMP or Photoshop. Websites such as mybannermaker.com can help you make banners, but you won't be able to make a good quality banner like what Neo has in her signature.
To put a banner on Rescreatu, you need to upload it to an image hosting site such as tinypic.com, and then get the HTML code in order to put it in your signature or on your profile.
There's no certain size it has to be, but keep in mind for signatures they can't exceed about 500px? If any bigger they will likely stretch the page and that is quite annoying and unpleasant.
I actually think it's 600px+ for width. So the max size you should risk is probably around 600px width by 300px height. Remember also that tall signatures can be obnoxious as well.