The two most common faults with old CD players are: 1) laser failure, either out of alignment, dirty or servo problems preventing movement; all exhibiting failure to read disc as symptoms. 2) Mechanical tray failure, preventing a disc from being inserted or extracted from the player. The former maybe fixable with a replacement laser, but these can be hard to source and not always straight forward to fit and align. Broken trays can be as simple a belts popping off to cog teeth worn and broken plastic casing; sometimes easy fixes, other time difficult to repair.
Also note, many early CD players are unable to play CD-R or CD-RW discs and often have rudimentary older DAC chips that can’t provide the audio quality of newer players. However a good CD player will have SPDIF output for connection to an external DAC, which can be a good way to greatly improve the performance of a basic CD player.