I love my nappy bag. But you can definitely do with any bag, as long as you're organised. That could be buying a few small zippered pouch bags to organise things in (like nappy stuff in one, baby food in another, dummies in another), or even clear bags like zip locks would allow you to see what you're after.

It can be done, the great thing about nappy bags is that the compartments are already there, but you could definitely work around that. The only thing it wouldn't have would be a nice deep pocket to put a bottle/water bottle but I'm sure you could think of something.