Is this a bag for a gift? If so, the bag's paper should be at least twice the size of the gift itself so that the gift will easily fit inside the bag. The paper you use for this gift only has to be sturdy enough to last until the gift is opened. If you're making a paper bag that you want to use regularly (or at least more than once or twice), then this should be taken into account when choosing the paper. Newspaper will usually break down pretty quickly in individual sheets. Thicker, heavier stock paper--like paper the thickness of computer paper or magazine pages--won't break down nearly as quickly and will help the paper bag last longer.
Beyond choosing the paper for durability, do you want a certain look to the bag? As the paper will most likely be recycled from something else and will most likely be used for a gift, maybe the homemade bag should have a theme to it. If the recipient of the gift to is a sports fan, then maybe use the sports sections from a lot of different exciting days in sports to make the bag. If you're a nature buff, then magazine paper from a National Geographic might be the way to go, so that the photos visible on those pages give the bag a unique style and design.
Actually making the bag can be done in some simple folding and crafts steps and is best done when making homemade paper bags as presents for people. On one end, fold down your paper a couple of inches to make the top of the bag. Make sure these are firm creases. Set your box on top of the paper folds. Bring the paper around the box and tape it, as if you were about to wrap it normally. At the unfolded end of the paper, fold in the flaps like you would normally and tape or staple, whichever you think works better. At the top of the bag, punch holes for a handle made out of ribbon, or tape a homemade handle of twisted cardboard inside the sides of the bag itself. Cutting a piece of cardboard to glue or tape onto the bottom of the bag to give it stability and strength is also a good idea.