The best thing is to designate a specific amount of space and stay within that. Whether it is a single box or an entire basement! There is nothing at all wrong with collecting textiles. I love and covet fabric of all sorts. Other people let the stores "store" it for them. But you quickly find out that if you find a piece of fabric you fall in love with and don't get it, that even if you go back the next day the likelihood is it will all be gone and you will never find it again...
So ask yourself first what/why do you wish to keep your scraps? I have a scrappy style and that is fine for me, but if you don't want to do projects like that why accumulate bits you won't ever really use?? Many of us have a thrifty nature and it bugs us to throw things away, but you can also give them away or even sell them if you must.
When I first started, I threw my scraps into a hamper but that didn't work well for me. It was a horrible mess, everything needed to be ironed all the time, you had to dump the whole thing out and dig through it to find anything. I prefer to use boxes now which are sorted either by color groups or types. So I have boxes for blue, red, brown, etc. as well as "farm", batik, "travel". And in the last couple of years I decided that while yes, I needed a lot of different fabrics for my style of work, I didn't need a lot of yardage or a lot of different sizes of small pieces to take care of, so now I don't keep anything smaller than a 6.5" strip, and I've given away all my smaller pieces. I can always cut a 2" square from a 6.5" strip if I need to and I don't need to deal with the remainders.
As a piece of yardage gets smaller than a fat quarter, a strip is cut off, put in the box, and any usable rest (or if it is too small) goes to a crumb quilter I met here on the boards. I keep a box by my cutting area and send it to her when it is full.
The short form is that you decide how much space you wish to devote to stash, and how big of pieces you wish to keep in that stash. Keep that in mind for when you see fabric that you wish to buy. I rarely buy anything larger than a 3 yard piece, mostly 1/2 yard cuts. I have a friend who rarely buys anything smaller than a 6 yard piece and will happily buy a bolt (typically 8-14 yards) if she loves it.