Yes. You just need to find a "social" (which is only to say, "not HNMLS") shell that has both the right attitude and a critical mass of members. (I mention the 2d part because a lot of fine shells have had their share of retirements, etc.)
My own shell is always doing these sorts of things. Sunday, for example, was a massive COP4.2 (Ouryu/ The Savage) run with 11 clears (in 5 fights, mind you). A week ago it was COP3.5 for many of the same group. A few weeks ago it was a "Promy-thon." About two months ago we did a series of events to "get folks to Sky." And sometimes it's even "smaller" things like limit-breaks, helping with keys and coffers, and so on.
I do not want to tear people away from their LS families, but if you really despair of doing such things in your current situation, visit cynosure-online.org (/forums too, but don't skip the front page, which is a scrapbook of sorts).
____________________________Grommet Taru male San d'Orian
95 WHM, BLM, RNG, SMN, WAR/ 64SAM/ 52DRK/ 51PLD, THF/ 49NIN, SCH, RDM, DNC
100 Woodworking and Fishing; 80 Synergy
60 in all other crafts
(Updated November 12, 2011)