Is it possible to build homemade storage device?
Yes, you can do it-- but it's hard and it won't store very much.  I think the thing that makes it hard is that you need to know a lot of very specialized areas to make it works. Things like:  software, signal processing, electronics, electro-magnetism, metalworking, motors/gears/etc, and materials science (somewhat like chemistry).  While not impossible, it is rare to find someone who is proficient in all of those areas.  

If you want to make this easier, I would recommend starting with a standard cassette tape player/recorder.  Rip out all of the electronics and keep the motors, gears, mechanical stuff, and the read/write/erase heads.  Then add back in your own electronics.  This still offers a lot of challenges, but the odds of success go way up. Then, if you get that working you can take the knowledge you gained and go on to a hard drive or something.

If you go the cassette tape route, let me mention that if you Google that you'll find a lot of pages that do something similar, but without modifying the player much.  They do that by modulating the data into something that resembles audio and can be stored as audio.  That's not what I'm recommending. If you rip out the guts of the player/recorder then you can have direct control of the motors and heads, which opens up lots of possibilities.

Hard drives will be harder, mostly because you'd have to figure out how to make the hard drive platters.  Meaning, you have to make the magnetic recording medium and somehow spread it out evenly and smoothly on the glass or aluminum platter "base".  Making the heads isn't easy either.

I should point out, however, that a clean room is not required.  I remember playing with a "removable hard drive" on a DEC PDP-8 computer.  Instead of removing the whole drive, you only removed the platters.  The platters were about 12 inches across and contained in something like a piece of Tupperware that you'd carry a cake in.  About 6 or so platters per carrier.  Before putting the platters into the drive you would have to remove them from the Tupperware.  It was big, and didn't store a lot, but no clean room either.  Don't get me wrong, modern drives do need a clean room.  But a DIYer has little or no hope of building a modern drive in his home so it's not really an issue.

Another form of storage that could be interesting is an optical fiber "drive".  Light travels approximately 6 inches per nanosecond in a fiber optic.  So if you have a fiber that is 100 feet long and you're transmitting stuff at 1 gbps then you're really storing 200 bits of data in that fiber.  Make the fiber several kilometers long and you could store a barely useful amount of data.  Get a fiber transmitter and receiver set up so whatever is received will be retransmitted and your data will just recirculate endlessly.  Some extra stuff will then allow you to read/write the data.

Probably the most useful, and least satisfying, thing to build would be something like a USB thumb drive.  Basically you buy the flash chip, and the controller chip, connect them together and you're done.  To make it a little harder, replace the controller chip with a microcontroller and write lots of software.  It's not super interesting, in my opinion.  I don't think it offers the same sense of accomplishment that the other approaches offer-- even though the performance and capacity would be the highest this way.