It is possible, but the complexity and size depends on what you call basic electronic components. An ALU and sequencer logic is a bit complex, but doable. Memory is simple, but the basic pattern must be repeated an very large number of times (think 1000's of times).
Besides the hardware you will also need software that runs on it. As a rough estimate, for a moderately complex CPU (classic 16-bit instructions 8-bit data level) your software effort will be comparable to your hardware effort. (For a more simplistic CPU you'll need more SW effort.) And how are you going to load that SW into your machine?
Bug hunting (and solving) will be an interesting undertaking. I'd advise you to start wrting in VHDL and run that on a simulator, that will be far easier to debug than a bunch of chips and wires.
Two students of mine created a 16-bit CPU with some basic software (including a GCC backend port) in ~ 1 year, starting with VHDL and C code for simulation. The ALU used 74181 chips, memory was static RAM, and they used an atMega to interface between the PC and their computer. The computer was partly on solderless breadbords and partly on PCBs (the 8 16 bit registers). (These two were not average students!)
