A full-grown house mouse does not need a very large hole to make his way into your house. To prevent mice entering your home make sure that all holes are stuffed with steel wool or copper mesh, or filled with caulk or plaster. Holes around the foundation of the home, and those between the house and garage are particularly important. Seal gaps around doors, and install floor sweeps, caulk openings around water pipes, electric wires and vents. Keeping food in sealed containers, not leaving dirty dishes out, cleaning up spills immediately, rinsing out cans and bottles before putting them in the recycling, and keeping a tight lid on your trash bin will reduce food availability. Clearing pet food away before you go to sleep is also advisable.
There are a variety of non-chemical means of dealing with mice, the most favourable being "live" traps which you can use to capture and release mice (well away from your home). These can be purchased at local hardware stores. Another option is a repellent sound device which, when in operation, will produce a high frequency sound (not heard by humans) that will annoy the mice (though there is some debate about the effectiveness of this option). If you decide to opt for a more "biological" method of control, remember that female cats are more predacious than males!
For more information on dealing with mice check out these websites:
Beyond
Pesticides (click on "Info Services - Alternatives Factsheets")
NCAP
Alternatives Factsheets