It is definitely possible to teach yourself. I can attest to this as I have no teacher(never have)! I've been playing for a year(and some months) but I think I have good technique because everything I can play is relatively easy
to me(Good technique = easy playing right?)

. As of now, my repertory consists of 12 pieces of progressive difficulty all above level 3 in the ABSRM grading system.
Some of the pieces I've accomplished(That I feel most of the forum would know)(In no particular order):
1.Arabesque 1 - Debussy
2.Valse Op.64 No.2 - Chopin
3.Valse Op.69 No.1 - Chopin
That's all of the classics that I know(I'm learning 4 more as of now). All of the other pieces in my repertory are modern.
Some examples(So you can look them up, they may interest you)

:
1.Besaid Island from FFXPC(Final Fantasy Piano Collections) - Nobuo Uematsu
2. Twilight Shore - Joe Hisaishi
3. A Sign of Hope(My first piece) - Nobuo Uematsu
4. Circle of Eternity - Yasunori Mitsuda
That's just to name a few. Here's a few words of advice. First the obvious...Never give up! Second, You'll need a LOT of motivation(listening to a lot of music you like can accomplish this). Third, get as much information as you can(Forums, Google, Youtube, talking to other musicians whenever possible)! Also, it would be very helpful to be able to read sheet music(I assume you already can). Self teaching is definitely a possible goal so go for it!