Having a giant electrical machine just for buffing metal isn't convenient for me. If I could, I would prefer to just sand away. Is it possible to obtain as good of a finish as you would see with a buffer by using continuously less course sand-paper layer by layer?
No, you definitely don't need a buffer
This gold belt buckle dates from the 6th century - long before buffing machines were invented. Finishing jewellery by hand is the traditional way to do it, and as you can see the results can be excellent.
However, you can't get this kind of finish just with sandpaper
Going up through the grades of sandpaper is a good way to start, but to achieve a truly smooth finish you need something finer. There are a few different things that are available now for final polishing, but the traditional one is ferric oxide, also known as jeweller's rouge.
You can use this by applying a little to a polishing cloth, and polishing the piece until it's done (you'll need to wipe the rouge off afterwards)
Using a Dremel or Rotron tool with a buffing or polishing wheel mounted will radically shorten the time needed to get a high polish on any surface.
If you mean to limit yourself to non-mechanical methods then jewelers polish, diamond paste, or aluminum oxide pastes are what you need .. applied to a sponge, cloth, or bit of soft leather.