Can I get all of the evolutions of eevee in just one eevee?

No, you can't unevolve any Pokemon, and none of the eeveelutions can evolve twice.

BUT-- you can take that one eevee to the breeders in Goldenrod (South end, building next to the night watchman)
Give your eevee to one of the people (doesn't matter which) and give a ditto to the other. You can then go out into the little yard and visit your Pokemon. When you talk to your eevee, it should say "shows interest in ditto" If it's not showing any interest, then they're not going to breed, but ditto is able to breed with any Pokemon so it should work. Come back in a day or two and the old man will be standing in the yard with your Pokemon. Talk to him and he will give you an egg.
Now comes the fun part! Eevee egg take the longest to hatch, something like 9,000 steps. So walk EVERYWHERE. You can check the stats of your egg to see how close it is to hatching. It will rather say "needs more time" or "it's wiggling around inside" and the egg sprite (animation) will wiggle. From there it's maybe about 1000 more steps (my guess) When it finally hatches, you'll have a level 5 eevee.

Also, if you breed your eeveelutions together, your baby eevee could inherit some of their moves. Example-- I bred my lvl 80 male umbreon and my lvl 35 female espeon and my eevee was born with return (from umbreon) an iron tail (from both)

I realize this is a question and not an answer but I'll elaborate on this other question. Does eevee's evolutions evolve? Like for example does umboreon evolve to an even better Pokemon? Or does leafeon evolve? I've been wondering for the longest time and I just want to know if there is a final evolution after an eevee's first evolution. Help!