That is a comfortable space to build a J-tube inside. You would want to use firebricks (4 1/2" x 9" x 2 1/2"). As you say your house is not big, I presume you are thinking of a 6" system. You want the core to be 6" square inside for
feed, burn tunnel and riser. This would require cutting some firebricks, though you could make the sides two bricks high and add brick pieces (from the old liner material?) to the floor to bring the internal height to 6". You could run a single firebrick row down the middle of the floor and fill beside it with clay, which would fire hard soon. Filling around the brick core with cob would stabilize everything.
You could actually make just the feed tube which sees the most abrasion from brick, and form everything else from cob. As long as it doesn't get scraped vigorously, it will last a long time. Making a 6" square box from scrap
wood would make it easy to get the right size and smooth surfaces for burn tunnel walls and ceiling. You can do the same for the riser as long as you reinforce the outside of it with something like
chicken wire to keep cracks from opening.
Here is a similar one that I built.
As I mentioned, you will need to cut a hole in the back of the steel box top to clear the heat riser. I would make it 12" x 12" to give comfortable working space. You can mount a barrel over the riser, and cob around it for mass. You do want to make something removable to inspect the top of the riser, either a barrel lid or the whole barrel.
Running the exhaust 8' along the floor may be problematic, depending on how good the natural draft of your chimney is. Does it draw any air current in when the system is stone cold? If so, you are probably good. I would suggest making this pipe 8" to reduce friction, or even better half barrels laid flat, and encased in cob.