I have found a berry basket, the kind that strawberries are often sold in, to be a very convenient way to store pet toys. I keep a basket filled with toys on the floor so that my dogs can take out whatever toy they want. (They haven't learned to put the toys back though.)
The basket can easily be washed whenever I wash the toys. I have a second basket I keep in a cabinet that is filled with different toys. Switching toy baskets keeps the dogs from getting bored with their toys.