Do I need a bike bag / box to put my bike on the Eurostar?
That page actually lays it out pretty well. You have three options: box/bag your bike, pay £30 for a reserved bike space, or pay £22 to ship your bike such that it will arrive within 24 hours of you.

The reserved bike spaces and shipping don't require you to box or bag your bike.

When I traveled on EuroStar and on French trains, I had a friend handy with a sewing machine make me a housse out of lightweight ripstop nylon. By turning the handlebars sideways and removing the pedals and wheels I was able to get it mostly within the size restrictions - it was bulky but fit in the luggage area, which is all that matters. It was slightly longer and wider than allowed but no one checked.