Leftover native grass seed should be kept in a constant environment. Constant temperature and humidity. Seed storage experiments in Kansas and Nebraska showed that seed kept at a constant temperature and humidity maintained its viability for many years. In some cases it actually was better in the second and third year after harvest when properly stored.
I doubt that many of you have an environment chamber where you can control both temperature and humidity. I recommend that you store your seed where there will be the least fluctuation in the elements. It is my experience that humidity is more critical than temperature, with the exception of extreme temperatures. Temperatures above 100 should be avoided.

Seed that you plan to plant next year can safely be kept in your basement or garage in trash containers. The main concern is to keep the seed from getting wet.

If you are going to store it over the summer it is best to keep it in a basement or other similar area. The extreme fluctuations of temperature and humidity in summer will affect the quality.

Don't store your seed in your attic or hayloft of your barn. If you have stored seed several years old it is best to have a germination test done before you plant it. Your local Extension office can recommend a reputable seed testing lab in your area.