I just use felt from Spotlight and other craft and $2 stores.
I glue it directly into place - ie not onto cardboard first - using the blue solid lipstick style dispenser glue that you will find in newsagents for sticking paper and cardboard. I have had no failures over the three years that I have been using this technique. When I oil the box prior to fitting the felt, I make sure that the areas where I intend to use the glue are masked so that no oil covers those areas.
There are usually two types of felt to be found in these places. Some are quite firm and some are quite soft.
The firmer stuff is easier to cut with scissors or knife but that is not really an issue with the softer stuff either because I stick a strip of blue painters' masking tape along where the cut will be and it is on that tape that I will mark the cut line with a ruler and pencil. I then use scissors to cut along that line and then I pull off the tape which only has a weak adhesive and pulls away easily without damage. Without using the tape, a straight, firm cut is a lot harder to do
The benefit of using the softer felt is that you can stretch it pretty easily to fit exactly into the corners and edges of your box. I usually just sharpen up an icypole stick to use to push the felt into the corners and edges. You can buy bags of those sticks in any craft store or Bunnings. You don't have to use them but that is what I have at hand.
The other benefit of using the softer felt is that there are a whole lot more colours available in the softer felts than in the stiffer variety.
There are plenty of things that men go into Spotlight to buy. They do have an amazing array of rectangular pieces of coloured felt in their craft area.
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