You can phone a bike shop and arrange to pick up a box from them: they're usually pleased to get rid of them because they have to pay to have them taken away. You can fold up the box and sling it across your shoulders then ride to the airport and spend a happy half hour dismantling the bike there.
In extremis you can go to a supermarket and nick a load of cardboard boxes and buy parcel tape and wrap the bike in the boxes, put the outers with the colourful designs on the inside, and it looks like a box.
Be aware that not a single cardboard box in the world will ever properly protect the bike so what the airlines want is protection for other people's luggage.
Be careful when acquiring one from a bike shop: you probably need a road bike sized box not a mountain bike one. At Madrid airport Iberia will insist on a box and you can buy one from them for €25, if they have any left! DAHIKT
You can also use clingfilm: I've done this, take off the handlebars, rear mech, front wheel, it's a bit complicated but doable. You can also go to a DIY store and buy strong plastic sheeting. It then looks like a bag and most airlines want a box or bag.