If you do in fact want a felt type fabric, and a thin fabric, go to your local pool table repair outlet and get pool table felt. Likely available online, also. As Joe mentioned, wool attracts moths, so a wool fabric would not be recommended. If this is your option, check out the Youtube videos for methods of applying pool table felt, these may help with adhesives to use.... similarly as applying other types of fabric, when laying out, start in the middle and work towards the edges. Don't pull/stretch your fabric too much when applying, it may distort and cause ripples. Somewhat similar to rolling out air pockets from under laminates or veneers, work from the center outward and don't stretch your fabric.
Sonny