All my scraps get sorted into two areas. Large pieces (larger than 4" wide) and patterned paper scraps get put into colour sorted drawers with embellishments of the same colour family. I sort pretty much everything by colour with the exception of a few themes (love, baby, Christmas and Halloween/Fall). I have a stack of 12x12 Iris drawers dedicated to all the colours of the rainbow plus brown, black/grey and white/cream. I also keep a drawer for true multi-coloured scraps and metallic and specialty papers as well as the above mentioned themes.
Smaller scraps are put into photo box ready to be die cut or used in projects as required. Generally, when I've got some free moments but aren't feeling particularly creative, I'll take out the box and die cut some Nestabilities or Quickutz shapes for future projects.
I keep my diecuts in photo keepers inside another photo box. Each is labelled with the type of shape it is so I can easily find a scalloped circle or square if I need it.
I think organizational style is pretty personal but one product that I can highly recommend no matter what your personal style is a label maker. I get so much use out of my label maker - I actually wore one out and had to get another one...
If you have a more modest pile of scraps you might also consider an accordian file (make sure its photo safe) or use file folders in a filing cabinet to store scraps by colour.
So what do you do if you run out of space in your scrap system...get rid of your scraps! Honestly, once you start to out grow your chosen system its either time to get a new system or time to let go of some of your scrap paper. If you have die cuts sell them on EBay - or better yet, send a nice RAK to someone who will really appreciate them. There are lots of folks out there without access to nestabilities or the big Marvy punches who would love to open an envelope filled with multicoloured scalloped circles!
But use your scraps whenever you can. Let's face it, paper crafting isn't the most environmentally friendly hobby, so re-use and recycle wherever you can. I find that even the smallest scrap can be used with a tiny punch or to diecut letters for cards and scrapbook layout titles.
Here is box 3 in my series on boxes, bags and gifts. This is a 2-5-7-10 box with flap and you can find the instructions for making your own version here. I used a large wheel stamp to make the background and then punched flowers from double sided adhesive paper and adhered them to the cardstock. I then applied glitter to the flowers after removing the top layer of adhesive. Inside the bag is a "homemade" glitter kit with mini bags of glitter and punched or diecut images from double sided adhesive.