I'm starting a project where I need to cut a 114mm diameter, 33mm deep hole into a piece of wood. My plan is to keep the circular piece, then to sand it. My question is, what would be the best piece of equipment to use in order to make a clean. straight cut?
This can probably be done with 4.5" hole saw which is pretty close to your dimensions.
I assume what you want is the disk, not the hole.
A circular disk can also be cut with a scroll saw.
Another tool is an adjustable circle cutter or fly cutter of which there are few different types.
A circle can also be cut using a mill (or lathe) with a milling cutter in combination with a rotary table (or divider).
While a 4 1/2 inch hole saw will do the trick it would be an expensive investment if this is a one time project. The hole saw will leave a center hole from the pilot drill. You may be able to limit this to one side of the disk if you removed the pilot drill just prior to finishing the cut. You will also need a hefty drill with a 1/2 inch chuck. I would suggest either using 2 thinner boards and gluing them together prior to sanding that way you can use an relatively inexpensive jigsaw or try it with a quality jigsaw going slowly with a long blade. If you must use a single full thickness board then as @Tyler Durden suggested a scroll saw.
I think you can do this with a router. Create a template in some thin material using one of the methods above. Hole saw, circle cutter, jigsaw, even a hand saw, etc. Sand it until it is as perfect as you can get it. Then somehow tack it down to your 33mm stock (nails or screws), and use a flush cutting router bit with pilot bearing to cut the thick stock. Here are some router bits: http://www.mlcswoodworking.com/shopsite_sc/store/html/smarthtml/pages/bt_flush.html See number 7805, for example.