These are good questions. My short answer is $300.
My regular Omaha game is $6/$12 with a one-third kill (to $8/$16). Let's assume that roughly a third of hands are killed. This means the average stake in my game is $6.67/$13.33, and the average stake in your game ($5/$10 with a full kill) is the same.
So you could say $13.33 x 25 big bets = $333. But wait. If you play very tight preflop, Omaha has less variance (and therefore requires a lower bankroll) than LHE at the same stakes. And in my experience, $300 in my $6/$12 Omaha game lasts about as long as $300 in a similarly loose $6/$12 Limit Hold'em game.
Mind you, I
don't play very tight preflop. I usually buy in for $300, and it's not uncommon for me to be in a game $600 or more. Yesterday afternoon I was stuck $400 after about 2 hours and didn't feel I was on tilt or had made any big mistakes. Three and a half hours later, I cashed out up $400. That gives you an idea of the swings in my game and with my playing style.
Now let me quote Steve Badger on relative bankroll requirements: "Omaha is a great game to make money if you have a small bankroll. $3/6 Omaha should require less of a bankroll for a sensible player than $3/6 Limit Hold'em, but generate a higher hourly win rate." That's from his extremely helpful introduction to the game, which you can find at
http://www.stevebadger.com/poker/omaha/strategy/.
What he wrote confirms your perception that your stacks are dwindling far more slowly. Why is this happening? Because even though you're not scooping often, you're picking up some half pots and quarter pots along the way. You're not dragging as many chips
per pot won as you would in LHE, but you're dragging chips more often—and just as many chips overall—which means less variance and lower bankroll requirements.
Finally, I want to recommend to you the Hutchison point count system for starting hands. Other forum members might say this is an oversimplification and a crutch and that a thinking player would never use it. But if you're new to the game and trying to play really tight preflop, it can be a helpful starting point:
http://ehutchison.homestead.com/OmahaSystem.html.
Good luck!