Eventful (and similar sites) are pretty involved sites. It's hard to really give much feedback since there's still a matter of scope here (you could probably throw one together in a day, but you could also spend months doing one, too).
My suggestion is to break the project down. That helps you out a couple different ways: you get a better handle of what your timeframe should be overall, but secondly you really start understanding what specifically you're going to have to be attacking here. 37signals suggests the same thing:
It's hard to just say "I want site x, now how long should I spend on it?" But if you break things down into smaller pieces it becomes easier to estimate, particularly since you have that prior Rails experience.
If you take an iterative approach to development then I see no reason why you couldn't have an initial prototype/beta release out in a month or two - but depending on how you want to develop its functionality it could take anywhere from 2/3 months to a year.
As I prefer an agile approach to development, I'd say list out the core functionality you want your app to have. In XP and other agile methods you might call these "User stories" - think of these as light-weight use cases, i.e.:
"As a user I want to see upcoming events that my friends are attending, so that I can arrange to meet up with them."
Most user stories will take a similar form "As xxx I want to do yyy so that I can xxx". If you find your self struggling to fill in "zzz" (i.e. a good reason for a function) then maybe you don't really need it?
After that, split up your stories into "Essential", "Nice to have", and "If I have time" piles and work through them in fixed length iterations (say 1/2 weeks long) with the Essential pile always taking priority. Schedule in the stories you want to tackle at the beginning of each iteration and only tackle those. If you get a new idea for a feature, write a new story card ready to be planned into a future iteration.
I find this helps keep development on track, keeps you focussed on essential functionality and helps you get releases out early and often.
I think 14 weeks is more than enough time. You should probably work with someone though, to prevent burnout, and to stay focused.
We built a similar project, basic profile page, adding friends, e-mailing in just under 4 weeks, but that was really the bare skeleton of the site and it was two of us. We didn't try things like poking but we paced ourselves to 8 our days 7 days a week.
Good luck with the app.
ps: we also used RoR, would have been 10x long if we hadn't.