If you knit from the bottom up (short length on the needles) you will have to have a separate length of yarn for each stripe.
If you knit lengthwise, you can knit with one color at a time, changing when your stripe is as wide as you want it.
The lengthwise one will be easier in that you won't have to worry about tangling yarn or getting holes between the colors. You'll just have to cast on lots and lots of stitches and each row will be really long. It'll be the same number of stitches, either way.
If it's a really long scarf, you might want to consider using two circular needles as if they were two straight needles. You could put point protectors or rubberbands on the end of each to keep the stitches from falling off. It might be too bunched up on one needle to knit it comfortably.