Eurostar

The chunnel train doesn't check luggage. You'll have to carry everything on the train yourself. There are racks above the seat for carry-on sized bags, and racks at both ends of the cars for larger bags.

When you board the Eurostar you take your luggage on with you. There will be racks that you can put your luggage on. Congrats, on calling the Eurostar by it's proper name. It isn't the chunnel, if you use that word in the UK you will get some strange looks.

Yes as Robin says, the EUROSTAR is self serve luggage, you carry it on, you store it , and you carry it off.
There are luggage areas at ends of cars. I tend to pick seats within eyesight but not always, its pretty low risk for someone walking off with your bag as its not like some milk run train with stops etc..

More than you probably need to know about Eurostar luggage, from the horse's mouth, aka the official site: http://www.eurostar.com/UK/uk/leisure/travel_information/at_the_station/baggage.jsp More info and links to resources, from The Man in Seat 61, including links to his Eurostar seating tips and seat charts:
http://seat61.com/London-to-Paris-by-train.htm#.UP037Ib4JSA (Sorry, the system here won't let me create a live link with the second URL. I tried.)

Thanks for all the info, the luggage storage and above racks is great info and appreciated

Another small tip: when boarding is called, try to get to your specified carriage ahead of the mob because the luggage space at the carriage end fills up very quickly. Once we were taking the Eurostar in the wintertime from London to Paris and a couple of yummy mummies with posh accents and several small children filled up the entire luggage space with skis, suitcases, backpacks, boots, etc., spilling out into the passageway making it almost impossible to pass by. (They also installed their children in seats that belonged to passengers who boarded at Ashford, but that's another story.)

Here's a live link for Suz's URL. How did I do it? Use Tiny URL to create a new, short URL, which lacks the characters such as #, %, and ? that this Helpline will not allow in live links.

Thanks, Harold. Great idea. Which obviously did not occur to me at the early morning hour when I was posting above. Hope I remember it for the next time! (I often use bit.ly instead of Tiny URL, which is why it's kind of embarrassing that it didn't occur to me to use bit.ly the other morning.)