Does the size of my snake matter?

Length is beneficial, girth is a hindrance. They both scale together though.

Length allows you to encircle and kill smaller snakes by forcing them into a small space they cannot escape from.

Girth reduces your 'turning circle'. Although you can turn just as quickly as other snakes, your freedom of movement is restricted - a common way for larger snakes to die is if they have two snakes either side of their head, leaving them no way to turn out of danger.

If you want to stay safe in the late-game (20k+ in size), the only way to reliably do so is to move in a large circle, following your own tail, so you always have one side of you clear to turn into if someone tries to attack you. Your rate of growth is relatively slow, you can only eat what you happen to go past in your big circular route, and you can't stray out too far from your own path without getting into danger, but if you are conservative in your risk-taking you can grow to essentially any size if you're careful with this strategy.