I suppose it depends on how you define "from scratch". Pretty much anyone can build a desktop PC from parts in a single day. That's what would make the most sense in this context as well, probably. Target a simple CPU design with a simple instruction set that you can understand.
If you want to go further from scratch... Designing a motherboard wouldn't be too hard, though it probably wouldn't be very good or reliable. Same with a power supply. Designing my own CPU is something I could probably do with several months' worth of effort, as a CompE. But actually making a physical chip of it is another matter entirely. It would be expensive and you'd have to pay a manufacturer to do it. Physically making your own CPU at home is pretty much out of the question, but you could use an FPGA. I have no idea how to go about making RAM or hard drives "from scratch". Same for monitors.
If you want to go further from scratch... Designing a motherboard wouldn't be too hard, though it probably wouldn't be very good or reliable. Same with a power supply. Designing my own CPU is something I could probably do with several months' worth of effort, as a CompE. But actually making a physical chip of it is another matter entirely. It would be expensive and you'd have to pay a manufacturer to do it. Physically making your own CPU at home is pretty much out of the question, but you could use an FPGA. I have no idea how to go about making RAM or hard drives "from scratch". Same for monitors.