Yep, Marda is right -- five month olds do not have tantrums yet; they have needs that are not being met (they're hungry, they're tired, they hate the way something in the car seat is poking them, they are startled by being iin the stroller and in unfamilar surroundings). Please don't imagine she's intentionally being naughty and resisting to misbehave because at five months she simply is not developmentally capable of that kind of thinkiing and manipulation. (Someone posted that you musn't let her think she's the boss. She's not capable of that kind of thinking yet!) And of course she wants to be carried. She's an infant and feels safest when physically close to you. Try not to expect more than she is capable of handling at this age.
At five months she needs some stimulation and yes, you should try short runs -- the nearest park and some time on a blanket on the ground, then home again, as someone noted. That's enough at her age.
Be sure too that when she's in the stroller she knows you are there -- Does her stroller have her facing away from you or toward you? If she's facing away from you in the stroller, you have vanished in her mind; get a stroller where she faces you and you might see a difference. To infants this young, when you are out of sight, you have disappeared permanently; they have zero idea that you actually exist when you're not with them, so her stroller distress after more than a short around-the-yard run may come from the fact that the stroller means mom is gone for good.
Also reassess: Are you thinking in terms of "let's go for long walks, go to the mall play area" etc? What did you have in mind by exploriing? She can get overstimulated too, remember. Make only short runs, have her where she can see you, time things so she is well fed and not close to a nap-- and naps are a lot of the time at this age. You have tons of time later for lots of great exploring. Right now, take it easier and try not to be frustrated with her because she's not capable yet of handling the outings you had in mind.